christopher/oeis · Datasets at Hugging Face

a-number stringlengths 7 7	sequence sequencelengths 1 377	description stringlengths 3 852
A000101	[ "3", "5", "11", "29", "97", "127", "541", "907", "1151", "1361", "9587", "15727", "19661", "31469", "156007", "360749", "370373", "492227", "1349651", "1357333", "2010881", "4652507", "17051887", "20831533", "47326913", "122164969", "189695893", "191913031", "387096383", "436273291", "1294268779" ]	Record gaps between primes (upper end) (compare A002386, which gives lower ends of these gaps).
A000102	[ "0", "0", "0", "0", "1", "2", "5", "12", "27", "59", "127", "269", "563", "1167", "2400", "4903", "9960", "20135", "40534", "81300", "162538", "324020", "644282", "1278152", "2530407", "5000178", "9863763", "19427976", "38211861", "75059535", "147263905", "288609341", "565047233", "1105229439", "2159947998" ]	a(n) = number of compositions of n in which the maximum part size is 4.
A000103	[ "0", "0", "1", "1", "2", "5", "12", "34", "130", "525", "2472", "12400", "65619", "357504", "1992985", "11284042", "64719885", "375126827", "2194439398", "12941995397", "76890024027", "459873914230", "2767364341936", "16747182732792" ]	Number of n-node triangulations of sphere in which every node has degree >= 4.
A000104	[ "1", "1", "1", "2", "5", "12", "35", "107", "363", "1248", "4460", "16094", "58937", "217117", "805475", "3001127", "11230003", "42161529", "158781106", "599563893", "2269506062", "8609442688", "32725637373", "124621833354", "475368834568", "1816103345752", "6948228104703", "26618671505989", "102102788362303" ]	Number of n-celled free polyominoes without holes.
A000105	[ "1", "1", "1", "2", "5", "12", "35", "108", "369", "1285", "4655", "17073", "63600", "238591", "901971", "3426576", "13079255", "50107909", "192622052", "742624232", "2870671950", "11123060678", "43191857688", "168047007728", "654999700403", "2557227044764", "9999088822075", "39153010938487", "153511100594603" ]	Number of free polyominoes (or square animals) with n cells.
A000106	[ "1", "2", "5", "12", "30", "74", "188", "478", "1235", "3214", "8450", "22370", "59676", "160140", "432237", "1172436", "3194870", "8741442", "24007045", "66154654", "182864692", "506909562", "1408854940", "3925075510", "10959698606", "30665337738", "85967279447", "241433975446", "679192039401", "1913681367936", "5399924120339" ]	2nd power of rooted tree enumerator; number of linear forests of 2 rooted trees.
A000107	[ "0", "1", "2", "5", "13", "35", "95", "262", "727", "2033", "5714", "16136", "45733", "130046", "370803", "1059838", "3035591", "8710736", "25036934", "72069134", "207727501", "599461094", "1731818878", "5008149658", "14496034714", "41993925955", "121747732406", "353221737526", "1025471857282", "2978995353959", "8658997820084" ]	Number of rooted trees with n nodes and a single labeled node; pointed rooted trees; vertebrates.
A000108	[ "1", "1", "2", "5", "14", "42", "132", "429", "1430", "4862", "16796", "58786", "208012", "742900", "2674440", "9694845", "35357670", "129644790", "477638700", "1767263190", "6564120420", "24466267020", "91482563640", "343059613650", "1289904147324", "4861946401452", "18367353072152", "69533550916004", "263747951750360", "1002242216651368", "3814986502092304" ]	Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!).
A000109	[ "1", "1", "1", "2", "5", "14", "50", "233", "1249", "7595", "49566", "339722", "2406841", "17490241", "129664753", "977526957", "7475907149", "57896349553", "453382272049", "3585853662949", "28615703421545" ]	Number of simplicial polyhedra with n vertices; simple planar graphs with n vertices and 3n-6 edges; maximal simple planar graphs with n vertices; planar triangulations with n vertices; triangulations of the sphere with n vertices; 3-connected cubic planar graphs on 2n-4 vertices.
A000110	[ "1", "1", "2", "5", "15", "52", "203", "877", "4140", "21147", "115975", "678570", "4213597", "27644437", "190899322", "1382958545", "10480142147", "82864869804", "682076806159", "5832742205057", "51724158235372", "474869816156751", "4506715738447323", "44152005855084346", "445958869294805289", "4638590332229999353", "49631246523618756274" ]	Bell or exponential numbers: number of ways to partition a set of n labeled elements.
A000111	[ "1", "1", "1", "2", "5", "16", "61", "272", "1385", "7936", "50521", "353792", "2702765", "22368256", "199360981", "1903757312", "19391512145", "209865342976", "2404879675441", "29088885112832", "370371188237525", "4951498053124096", "69348874393137901", "1015423886506852352", "15514534163557086905", "246921480190207983616", "4087072509293123892361" ]	Euler or up/down numbers: e.g.f. sec(x) + tan(x). Also for n >= 2, half the number of alternating permutations on n letters (A001250).
A000112	[ "1", "1", "2", "5", "16", "63", "318", "2045", "16999", "183231", "2567284", "46749427", "1104891746", "33823827452", "1338193159771", "68275077901156", "4483130665195087" ]	Number of partially ordered sets ("posets") with n unlabeled elements.
A000113	[ "1", "3", "4", "3", "6", "12", "8", "6", "4", "18", "12", "12", "14", "24", "24", "6", "18", "12", "20", "18", "32", "36", "24", "24", "30", "42", "12", "24", "30", "72", "32", "12", "48", "54", "48", "12", "38", "60", "56", "36", "42", "96", "44", "36", "24", "72", "48", "24", "56", "90", "72", "42", "54", "36", "72", "48", "80", "90", "60", "72", "62", "96", "32", "12", "84", "144", "68" ]	Number of transformation groups of order n.
A000114	[ "3", "4", "6", "12", "12", "24", "24", "36", "36", "60", "48", "84", "72", "96", "96", "144", "108", "180", "144", "192", "180", "264", "192", "300", "252", "324", "288", "420", "288", "480", "384", "480", "432", "576", "432", "684", "540", "672", "576", "840", "576", "924", "720", "864", "792", "1104", "768", "1176", "900", "1152", "1008", "1404", "972", "1440" ]	Number of cusps of principal congruence subgroup GAMMA^{hat}(n).
A000115	[ "1", "1", "2", "2", "3", "4", "5", "6", "7", "8", "10", "11", "13", "14", "16", "18", "20", "22", "24", "26", "29", "31", "34", "36", "39", "42", "45", "48", "51", "54", "58", "61", "65", "68", "72", "76", "80", "84", "88", "92", "97", "101", "106", "110", "115", "120", "125", "130", "135", "140", "146", "151", "157", "162", "168", "174", "180", "186", "192", "198", "205", "211", "218", "224", "231", "238" ]	Denumerants: Expansion of 1/((1-x)(1-x^2)(1-x^5)).
A000116	[ "1", "2", "4", "8", "20", "56", "180", "596", "2068", "7316", "26272", "95420", "349716", "1290872", "4794088", "17896832", "67110932", "252648992", "954444608", "3616828364", "13743921632", "52357746896", "199911300472", "764877836468", "2932031358484", "11258999739560", "43303843861744", "166799988689300" ]	Number of even sequences with period 2n (bisection of A000013).
A000117	[ "1", "2", "4", "8", "18", "44", "122", "362", "1162", "3914", "13648", "48734", "176906", "649532", "2405236", "8964800", "33588234", "126390032", "477353376", "1808676326", "6872485104", "26179922024", "99957747388", "382443112538", "1466024067850", "5629516646996", "21651955485304", "83400061453514" ]	Number of even sequences with period 2n (bisection of A000011).
A000118	[ "1", "8", "24", "32", "24", "48", "96", "64", "24", "104", "144", "96", "96", "112", "192", "192", "24", "144", "312", "160", "144", "256", "288", "192", "96", "248", "336", "320", "192", "240", "576", "256", "24", "384", "432", "384", "312", "304", "480", "448", "144", "336", "768", "352", "288", "624", "576", "384", "96", "456", "744", "576", "336", "432", "960", "576", "192" ]	Number of ways of writing n as a sum of 4 squares; also theta series of four-dimensional cubic lattice Z^4.
A000119	[ "1", "1", "1", "2", "1", "2", "2", "1", "3", "2", "2", "3", "1", "3", "3", "2", "4", "2", "3", "3", "1", "4", "3", "3", "5", "2", "4", "4", "2", "5", "3", "3", "4", "1", "4", "4", "3", "6", "3", "5", "5", "2", "6", "4", "4", "6", "2", "5", "5", "3", "6", "3", "4", "4", "1", "5", "4", "4", "7", "3", "6", "6", "3", "8", "5", "5", "7", "2", "6", "6", "4", "8", "4", "6", "6", "2", "7", "5", "5", "8", "3", "6", "6", "3", "7", "4", "4", "5", "1", "5", "5", "4", "8", "4", "7", "7", "3", "9", "6", "6", "9", "3", "8", "8", "5" ]	Number of representations of n as a sum of distinct Fibonacci numbers.
A000120	[ "0", "1", "1", "2", "1", "2", "2", "3", "1", "2", "2", "3", "2", "3", "3", "4", "1", "2", "2", "3", "2", "3", "3", "4", "2", "3", "3", "4", "3", "4", "4", "5", "1", "2", "2", "3", "2", "3", "3", "4", "2", "3", "3", "4", "3", "4", "4", "5", "2", "3", "3", "4", "3", "4", "4", "5", "3", "4", "4", "5", "4", "5", "5", "6", "1", "2", "2", "3", "2", "3", "3", "4", "2", "3", "3", "4", "3", "4", "4", "5", "2", "3", "3", "4", "3", "4", "4", "5", "3", "4", "4", "5", "4", "5", "5", "6", "2", "3", "3", "4", "3", "4", "4", "5", "3" ]	1's-counting sequence: number of 1's in binary expansion of n (or the binary weight of n).
A000121	[ "1", "2", "2", "3", "3", "3", "4", "3", "4", "5", "4", "5", "4", "4", "6", "5", "6", "6", "5", "6", "4", "5", "7", "6", "8", "7", "6", "8", "6", "7", "8", "6", "7", "5", "5", "8", "7", "9", "9", "8", "10", "7", "8", "10", "8", "10", "8", "7", "10", "8", "9", "9", "7", "8", "5", "6", "9", "8", "11", "10", "9", "12", "9", "11", "13", "10", "12", "9", "8", "12", "10", "12", "12", "10", "12", "8", "9", "12", "10", "13", "11", "9", "12", "9", "10", "11", "8", "9", "6", "6", "10", "9" ]	Number of representations of n as a sum of Fibonacci numbers (1 is allowed twice as a part).
A000122	[ "1", "2", "0", "0", "2", "0", "0", "0", "0", "2", "0", "0", "0", "0", "0", "0", "2", "0", "0", "0", "0", "0", "0", "0", "0", "2", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "2", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "2", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "2", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "2", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "2", "0", "0", "0", "0" ]	Expansion of Jacobi theta function theta_3(x) = Sum_{m =-oo..oo} x^(m^2) (number of integer solutions to k^2 = n).
A000123	[ "1", "2", "4", "6", "10", "14", "20", "26", "36", "46", "60", "74", "94", "114", "140", "166", "202", "238", "284", "330", "390", "450", "524", "598", "692", "786", "900", "1014", "1154", "1294", "1460", "1626", "1828", "2030", "2268", "2506", "2790", "3074", "3404", "3734", "4124", "4514", "4964", "5414", "5938", "6462", "7060", "7658", "8350", "9042", "9828" ]	Number of binary partitions: number of partitions of 2n into powers of 2.
A000124	[ "1", "2", "4", "7", "11", "16", "22", "29", "37", "46", "56", "67", "79", "92", "106", "121", "137", "154", "172", "191", "211", "232", "254", "277", "301", "326", "352", "379", "407", "436", "466", "497", "529", "562", "596", "631", "667", "704", "742", "781", "821", "862", "904", "947", "991", "1036", "1082", "1129", "1177", "1226", "1276", "1327", "1379" ]	Central polygonal numbers (the Lazy Caterer's sequence): n(n+1)/2 + 1; or, maximal number of pieces formed when slicing a pancake with n cuts.
A000125	[ "1", "2", "4", "8", "15", "26", "42", "64", "93", "130", "176", "232", "299", "378", "470", "576", "697", "834", "988", "1160", "1351", "1562", "1794", "2048", "2325", "2626", "2952", "3304", "3683", "4090", "4526", "4992", "5489", "6018", "6580", "7176", "7807", "8474", "9178", "9920", "10701", "11522", "12384", "13288", "14235", "15226" ]	Cake numbers: maximal number of pieces resulting from n planar cuts through a cube (or cake): C(n+1,3) + n + 1.
A000126	[ "1", "2", "4", "8", "15", "27", "47", "80", "134", "222", "365", "597", "973", "1582", "2568", "4164", "6747", "10927", "17691", "28636", "46346", "75002", "121369", "196393", "317785", "514202", "832012", "1346240", "2178279", "3524547", "5702855", "9227432", "14930318", "24157782", "39088133", "63245949" ]	A nonlinear binomial sum.
A000127	[ "1", "2", "4", "8", "16", "31", "57", "99", "163", "256", "386", "562", "794", "1093", "1471", "1941", "2517", "3214", "4048", "5036", "6196", "7547", "9109", "10903", "12951", "15276", "17902", "20854", "24158", "27841", "31931", "36457", "41449", "46938", "52956", "59536", "66712", "74519", "82993", "92171", "102091", "112792", "124314", "136698" ]	Maximal number of regions obtained by joining n points around a circle by straight lines. Also number of regions in 4-space formed by n-1 hyperplanes.
A000128	[ "1", "2", "4", "8", "16", "31", "58", "105", "185", "319", "541", "906", "1503", "2476", "4058", "6626", "10790", "17537", "28464", "46155", "74791", "121137", "196139", "317508", "513901", "831686", "1345888", "2177900", "3524140", "5702419", "9226966", "14929821", "24157253", "39087571", "63245353", "102333486" ]	A nonlinear binomial sum.
A000129	[ "0", "1", "2", "5", "12", "29", "70", "169", "408", "985", "2378", "5741", "13860", "33461", "80782", "195025", "470832", "1136689", "2744210", "6625109", "15994428", "38613965", "93222358", "225058681", "543339720", "1311738121", "3166815962", "7645370045", "18457556052", "44560482149", "107578520350", "259717522849" ]	Pell numbers: a(0) = 0, a(1) = 1; for n > 1, a(n) = 2*a(n-1) + a(n-2).
A000130	[ "0", "0", "1", "2", "5", "20", "115", "790", "6217", "55160", "545135", "5938490", "70686805", "912660508", "12702694075", "189579135710", "3019908731105", "51139445487680", "917345570926087", "17376071107513090", "346563420097249645", "7259714390232227300", "159352909727731210835", "3657569576966074846118" ]	One-half the number of permutations of length n with exactly 1 rising or falling successions.
A000131	[ "2", "5", "21", "61", "214", "669", "2240", "7330", "24695", "83257", "284928", "981079", "3410990", "11937328", "42075242", "149171958", "531866972", "1905842605", "6861162880", "24805692978", "90035940227", "327987890608", "1198853954688", "4395797189206", "16165195705544", "59609156824273", "220373268471398", "816677398144221" ]	Number of asymmetrical dissections of n-gon.
A000132	[ "1", "10", "40", "80", "90", "112", "240", "320", "200", "250", "560", "560", "400", "560", "800", "960", "730", "480", "1240", "1520", "752", "1120", "1840", "1600", "1200", "1210", "2000", "2240", "1600", "1680", "2720", "3200", "1480", "1440", "3680", "3040", "2250", "2800", "3280", "4160", "2800", "1920", "4320", "5040", "2800", "3472", "5920" ]	Number of ways of writing n as a sum of 5 squares.
A000133	[ "2", "5", "30", "2288", "67172352", "144115192303714304", "1329227995784915891206435945914040320", "226156424291633194186662080095093570364871077725232774230036394136943198208" ]	Number of Boolean functions of n variables.
A000134	[ "2", "6", "9", "12", "15", "18", "21", "24", "27", "31", "34", "37", "40", "43", "46", "49", "53", "56", "59", "62", "65", "68", "71", "75", "78", "81", "84", "87", "90", "93", "97", "100", "103", "106", "109", "112", "115", "119", "122", "125", "128", "131", "134", "137", "141", "144", "147", "150", "153", "156", "159", "163", "166", "169", "172", "175", "178", "181", "185", "188" ]	Positive zeros of Bessel function of order 0 rounded to nearest integer.
A000135	[ "1", "2", "6", "13", "24", "42", "73", "125", "204", "324", "511", "801", "1228", "1856", "2780", "4135", "6084", "8873", "12847", "18481", "26416", "37473", "52871", "74216", "103596", "143841", "198839", "273654", "374987", "511735", "695559", "941932", "1271139", "1709474", "2291195", "3061385", "4078152", "5416322" ]	Number of partitions into non-integral powers.
A000136	[ "1", "2", "6", "16", "50", "144", "462", "1392", "4536", "14060", "46310", "146376", "485914", "1557892", "5202690", "16861984", "56579196", "184940388", "622945970", "2050228360", "6927964218", "22930109884", "77692142980", "258360586368", "877395996200", "2929432171328", "9968202968958", "33396290888520", "113837957337750" ]	Number of ways of folding a strip of n labeled stamps.
A000137	[ "1", "2", "6", "18", "58", "186", "614", "2034", "6818", "22970", "77858", "264970", "905294", "3102434", "10661370", "36722642", "126752218", "438294018", "1518032598", "5265341314", "18286911130", "63586988434", "221342104842", "771235606050", "2689688538646", "9388096331642" ]	Series-parallel numbers.
A000138	[ "1", "1", "2", "6", "18", "90", "540", "3780", "31500", "283500", "2835000", "31185000", "372972600", "4848643800", "67881013200", "1018215198000", "16294848570000", "277012425690000", "4986223662420000", "94738249585980000", "1894745192712372000" ]	Expansion of e.g.f. exp(-x^4/4)/(1-x).
A000139	[ "2", "1", "2", "6", "22", "91", "408", "1938", "9614", "49335", "260130", "1402440", "7702632", "42975796", "243035536", "1390594458", "8038677054", "46892282815", "275750636070", "1633292229030", "9737153323590", "58392041019795", "352044769046880", "2132866978427640" ]	a(n) = 2(3n)! / ((2n+1)!(n+1)!).
A000140	[ "1", "1", "2", "6", "22", "101", "573", "3836", "29228", "250749", "2409581", "25598186", "296643390", "3727542188", "50626553988", "738680521142", "11501573822788", "190418421447330", "3344822488498265", "62119523114983224", "1214967840930909302", "24965661442811799655", "538134522243713149122" ]	Kendall-Mann numbers: the maximal number of inversions in a permutation on n letters is floor(n*(n-1)/4); a(n) is the number of permutations with this many inversions.
A000141	[ "1", "12", "60", "160", "252", "312", "544", "960", "1020", "876", "1560", "2400", "2080", "2040", "3264", "4160", "4092", "3480", "4380", "7200", "6552", "4608", "8160", "10560", "8224", "7812", "10200", "13120", "12480", "10104", "14144", "19200", "16380", "11520", "17400", "24960", "18396", "16440", "24480", "27200" ]	Number of ways of writing n as a sum of 6 squares.
A000142	[ "1", "1", "2", "6", "24", "120", "720", "5040", "40320", "362880", "3628800", "39916800", "479001600", "6227020800", "87178291200", "1307674368000", "20922789888000", "355687428096000", "6402373705728000", "121645100408832000", "2432902008176640000", "51090942171709440000", "1124000727777607680000" ]	Factorial numbers: n! = 1234...*n (order of symmetric group S_n, number of permutations of n letters).
A000143	[ "1", "16", "112", "448", "1136", "2016", "3136", "5504", "9328", "12112", "14112", "21312", "31808", "35168", "38528", "56448", "74864", "78624", "84784", "109760", "143136", "154112", "149184", "194688", "261184", "252016", "246176", "327040", "390784", "390240", "395136", "476672", "599152", "596736", "550368", "693504", "859952" ]	Number of ways of writing n as a sum of 8 squares.
A000144	[ "1", "20", "180", "960", "3380", "8424", "16320", "28800", "52020", "88660", "129064", "175680", "262080", "386920", "489600", "600960", "840500", "1137960", "1330420", "1563840", "2050344", "2611200", "2986560", "3358080", "4194240", "5318268", "5878440", "6299520", "7862400", "9619560" ]	Number of ways of writing n as a sum of 10 squares.
A000145	[ "1", "24", "264", "1760", "7944", "25872", "64416", "133056", "253704", "472760", "825264", "1297056", "1938336", "2963664", "4437312", "6091584", "8118024", "11368368", "15653352", "19822176", "24832944", "32826112", "42517728", "51425088", "61903776", "78146664", "98021616" ]	Number of ways of writing n as a sum of 12 squares.
A000146	[ "1", "1", "1", "1", "1", "1", "2", "-6", "56", "-528", "6193", "-86579", "1425518", "-27298230", "601580875", "-15116315766", "429614643062", "-13711655205087", "488332318973594", "-19296579341940067", "841693047573682616", "-40338071854059455412", "2115074863808199160561", "-120866265222965259346026" ]	From von Staudt-Clausen representation of Bernoulli numbers: a(n) = Bernoulli(2n) + Sum_{(p-1)\|2n} 1/p.
A000147	[ "0", "0", "0", "0", "0", "1", "2", "7", "14", "32", "58", "110", "187", "322", "519", "839", "1302", "2015", "3032", "4542", "6668", "9738", "14006", "20036", "28324", "39830", "55473", "76875", "105692", "144629", "196585", "266038", "357952", "479664", "639519", "849425", "1123191", "1479972", "1942284", "2540674", "3311415" ]	Number of trees of diameter 5.
A000148	[ "1", "2", "7", "15", "28", "45", "70", "100", "138", "183", "242", "310", "388", "481", "583", "701", "838", "984", "1152", "1337", "1535", "1757", "2001", "2262", "2545", "2855", "3183", "3540", "3926", "4335", "4770", "5233", "5728", "6248", "6801", "7388", "8005", "8658", "9345", "10064", "10824", "11620", "12452", "13324", "14236" ]	Number of partitions into non-integral powers.
A000149	[ "1", "2", "7", "20", "54", "148", "403", "1096", "2980", "8103", "22026", "59874", "162754", "442413", "1202604", "3269017", "8886110", "24154952", "65659969", "178482300", "485165195", "1318815734", "3584912846", "9744803446", "26489122129", "72004899337", "195729609428", "532048240601", "1446257064291" ]	a(n) = floor(e^n).
A000150	[ "0", "0", "1", "2", "7", "20", "66", "212", "715", "2424", "8398", "29372", "104006", "371384", "1337220", "4847208", "17678835", "64821680", "238819350", "883629164", "3282060210", "12233125112", "45741281820", "171529777432", "644952073662", "2430973096720", "9183676536076" ]	Number of dissections of an n-gon, rooted at an exterior edge, asymmetric with respect to that edge.
A000151	[ "1", "2", "7", "26", "107", "458", "2058", "9498", "44947", "216598", "1059952", "5251806", "26297238", "132856766", "676398395", "3466799104", "17873508798", "92630098886", "482292684506", "2521610175006", "13233573019372", "69687684810980", "368114512431638", "1950037285256658", "10357028326495097", "55140508518522726", "294219119815868952", "1573132563600386854", "8427354035116949486", "45226421721391554194" ]	Number of oriented rooted trees with n nodes. Also rooted trees with n nodes and 2-colored non-root nodes.
A000152	[ "1", "32", "480", "4480", "29152", "140736", "525952", "1580800", "3994080", "8945824", "18626112", "36714624", "67978880", "118156480", "197120256", "321692928", "509145568", "772845120", "1143441760", "1681379200", "2428524096", "3392205824", "4658843520", "6411152640" ]	Number of ways of writing n as a sum of 16 squares.
A000153	[ "0", "1", "2", "7", "32", "181", "1214", "9403", "82508", "808393", "8743994", "103459471", "1328953592", "18414450877", "273749755382", "4345634192131", "73362643649444", "1312349454922513", "24796092486996338", "493435697986613143", "10315043624498196944" ]	a(n) = na(n-1) + (n-2)a(n-2), with a(0) = 0, a(1) = 1.
A000154	[ "1", "1", "2", "7", "35", "228", "1834", "17382", "195866", "2487832", "35499576", "562356672", "9794156448", "186025364016", "3826961710272", "84775065603888", "2011929826983504" ]	Erroneous version of A003713.
A000155	[ "0", "1", "2", "7", "44", "361", "3654", "44207", "622552", "10005041", "180713290", "3624270839", "79914671748", "1921576392793", "50040900884366", "1403066801155039", "42142044935535536", "1349948504738292193", "45940391206037470098" ]	Nearest integer to modified Bessel function K_n(1).
A000156	[ "1", "48", "1104", "16192", "170064", "1362336", "8662720", "44981376", "195082320", "721175536", "2319457632", "6631997376", "17231109824", "41469483552", "93703589760", "200343312768", "407488018512", "793229226336", "1487286966928", "2697825744960", "4744779429216" ]	Number of ways of writing n as a sum of 24 squares.
A000157	[ "1", "2", "7", "111", "308063", "100126976263592", "131867858014413288241233435594064", "2804519150441879896741320496043335469582478995067528608051651559815168" ]	Number of Boolean functions of n variables.
A000158	[ "1", "2", "8", "19", "41", "78", "134", "218", "339", "506", "730", "1023", "1397", "1884", "2477", "3218", "4118", "5192", "6486", "8010", "9795", "11888", "14302", "17066", "20256", "23889", "27999", "32637", "37863", "43695", "50218", "57495", "65545", "74431", "84257", "95030", "106840", "119799", "133941", "149311", "166071" ]	Number of partitions into non-integral powers.
A000159	[ "2", "8", "20", "152", "994", "7888", "70152", "695760", "7603266", "90758872", "1174753372", "16386899368", "245046377410", "3910358788256", "66323124297872", "1191406991067168", "22596344660865282", "451208920617687720", "9461897733571886372", "207894669895136763704", "4776019866458134139042" ]	Coefficients of ménage hit polynomials.
A000160	[ "1", "2", "8", "21", "48", "99", "186", "330", "556", "895", "1397", "2107", "3097", "4459", "6264", "8644", "11760", "15742", "20790", "27128", "34993", "44664", "56473", "70784", "87995", "108564", "132970", "161828", "195686", "235274", "281349", "334682", "396202", "466849", "547712", "639935", "744716", "863443" ]	Number of partitions into non-integral powers.
A000161	[ "1", "1", "1", "0", "1", "1", "0", "0", "1", "1", "1", "0", "0", "1", "0", "0", "1", "1", "1", "0", "1", "0", "0", "0", "0", "2", "1", "0", "0", "1", "0", "0", "1", "0", "1", "0", "1", "1", "0", "0", "1", "1", "0", "0", "0", "1", "0", "0", "0", "1", "2", "0", "1", "1", "0", "0", "0", "0", "1", "0", "0", "1", "0", "0", "1", "2", "0", "0", "1", "0", "0", "0", "1", "1", "1", "0", "0", "0", "0", "0", "1", "1", "1", "0", "0", "2", "0", "0", "0", "1", "1", "0", "0", "0", "0", "0", "0", "1", "1", "0", "2", "1", "0", "0", "1", "0", "1", "0" ]	Number of partitions of n into 2 squares.
A000162	[ "1", "1", "2", "8", "29", "166", "1023", "6922", "48311", "346543", "2522522", "18598427", "138462649", "1039496297", "7859514470", "59795121480" ]	Number of 3-dimensional polyominoes (or polycubes) with n cells.
A000163	[ "2", "8", "34", "136", "538", "2080", "7970", "30224", "113874", "426520", "1590338", "5906640", "21866794", "80725280", "297292610", "1092539736", "4007553922", "14675748416", "53663197698", "195962433328", "714734393378", "2603994002280", "9477643739746", "34463730424720" ]	Series-parallel numbers.
A000164	[ "1", "1", "1", "1", "1", "1", "1", "0", "1", "2", "1", "1", "1", "1", "1", "0", "1", "2", "2", "1", "1", "1", "1", "0", "1", "2", "2", "2", "0", "2", "1", "0", "1", "2", "2", "1", "2", "1", "2", "0", "1", "3", "1", "1", "1", "2", "1", "0", "1", "2", "3", "2", "1", "2", "3", "0", "1", "2", "1", "2", "0", "2", "2", "0", "1", "3", "3", "1", "2", "2", "1", "0", "2", "2", "3", "2", "1", "2", "1", "0", "1", "4", "2", "2", "1", "2", "3", "0", "1", "4", "3", "1", "0", "1", "2", "0", "1", "2", "3", "3", "2", "4", "2", "0", "2" ]	Number of partitions of n into 3 squares (allowing part zero).
A000165	[ "1", "2", "8", "48", "384", "3840", "46080", "645120", "10321920", "185794560", "3715891200", "81749606400", "1961990553600", "51011754393600", "1428329123020800", "42849873690624000", "1371195958099968000", "46620662575398912000", "1678343852714360832000", "63777066403145711616000" ]	Double factorial of even numbers: (2n)!! = 2^n*n!.
A000166	[ "1", "0", "1", "2", "9", "44", "265", "1854", "14833", "133496", "1334961", "14684570", "176214841", "2290792932", "32071101049", "481066515734", "7697064251745", "130850092279664", "2355301661033953", "44750731559645106", "895014631192902121", "18795307255050944540", "413496759611120779881", "9510425471055777937262" ]	Subfactorial or rencontres numbers, or derangements: number of permutations of n elements with no fixed points.
A000167	[ "0", "0", "0", "1", "2", "9", "49", "306", "2188", "17810", "162482", "1642635", "18231462", "220420179", "2883693795", "40592133316", "611765693528", "9828843229764", "167702100599524", "3028466654021205", "57708568527002410", "1157199837194069405" ]	Nearest integer to modified Bessel function K_n(2).
A000168	[ "1", "2", "9", "54", "378", "2916", "24057", "208494", "1876446", "17399772", "165297834", "1602117468", "15792300756", "157923007560", "1598970451545", "16365932856990", "169114639522230", "1762352559231660", "18504701871932430", "195621134074714260", "2080697516976506220", "22254416920705240440", "239234981897581334730", "2583737804493878415084" ]	a(n) = 23^n(2n)!/(n!(n+2)!).
A000169	[ "1", "2", "9", "64", "625", "7776", "117649", "2097152", "43046721", "1000000000", "25937424601", "743008370688", "23298085122481", "793714773254144", "29192926025390625", "1152921504606846976", "48661191875666868481", "2185911559738696531968", "104127350297911241532841", "5242880000000000000000000" ]	Number of labeled rooted trees with n nodes: n^(n-1).
A000170	[ "1", "1", "0", "0", "2", "10", "4", "40", "92", "352", "724", "2680", "14200", "73712", "365596", "2279184", "14772512", "95815104", "666090624", "4968057848", "39029188884", "314666222712", "2691008701644", "24233937684440", "227514171973736", "2207893435808352", "22317699616364044", "234907967154122528" ]	Number of ways of placing n nonattacking queens on an n X n board.
A000171	[ "1", "0", "0", "1", "2", "0", "0", "10", "36", "0", "0", "720", "5600", "0", "0", "703760", "11220000", "0", "0", "9168331776", "293293716992", "0", "0", "1601371799340544", "102484848265030656", "0", "0", "3837878966366932639744", "491247277315343649710080", "0", "0" ]	Number of self-complementary graphs with n nodes.
A000172	[ "1", "2", "10", "56", "346", "2252", "15184", "104960", "739162", "5280932", "38165260", "278415920", "2046924400", "15148345760", "112738423360", "843126957056", "6332299624282", "47737325577620", "361077477684436", "2739270870994736", "20836827035351596" ]	Franel number a(n) = Sum_{k = 0..n} binomial(n,k)^3.
A000173	[ "30", "1482", "2418", "24180", "35238", "263820", "395730", "473298", "698130", "763620", "2212026", "2233554", "172459210", "209524210", "341354790", "384121920", "525150234", "530946330", "582129630", "1799281330", "2069510520", "2514290520", "3344596854", "5251502340" ]	Unitary-sociable numbers (smallest member of each cycle).
A000174	[ "1", "1", "1", "1", "2", "2", "1", "1", "2", "2", "2", "2", "2", "3", "2", "1", "3", "3", "3", "3", "4", "3", "3", "2", "2", "4", "4", "4", "4", "5", "3", "3", "4", "3", "5", "5", "5", "6", "5", "3", "5", "5", "5", "6", "5", "6", "5", "4", "4", "6", "7", "6", "8", "8", "7", "5", "6", "5", "8", "8", "4", "9", "7", "6", "7", "7", "8", "9", "9", "7", "9", "6", "7", "8", "10", "9", "9", "11", "7", "8", "8", "8", "11", "11", "9", "12", "11", "7", "9", "9", "12", "13", "9", "10", "11", "8", "6", "10", "13", "13", "13" ]	Number of partitions of n into 5 squares.
A000175	[ "1", "1", "2", "11", "38", "946", "4580", "202738", "3786092", "261868876", "1992367192", "2381255244240", "21411255538848", "2902625722978656", "451716954504285504", "319933105641374465472", "3761845343198709705600" ]	Related to zeros of Bessel function.
A000176	[ "2", "11", "46", "128", "272", "522", "904", "1408", "2160", "3154", "4306", "5888", "7888", "10012", "12888", "16384", "19680", "24354", "29866", "34816", "41888", "49778", "56744", "66816", "78000", "87358", "100602", "115712", "128112", "145804", "165712", "180224", "203040", "228964", "246932", "276480" ]	Generalized tangent numbers d_(n,2).
A000177	[ "1", "1", "1", "1", "2", "2", "2", "1", "2", "3", "2", "2", "3", "3", "3", "2", "3", "4", "4", "3", "5", "5", "4", "3", "4", "5", "5", "5", "5", "7", "6", "4", "5", "6", "6", "6", "9", "7", "8", "6", "6", "9", "7", "7", "9", "10", "8", "7", "7", "9", "10", "9", "11", "12", "13", "8", "10", "11", "10", "11", "11", "12", "13", "11", "9", "14", "13", "12", "16", "14", "14", "12", "13", "13", "15", "15", "15", "18", "16", "13", "14", "18", "15", "17", "20", "17", "21", "15", "15" ]	Number of partitions of n into 6 squares.
A000178	[ "1", "1", "2", "12", "288", "34560", "24883200", "125411328000", "5056584744960000", "1834933472251084800000", "6658606584104736522240000000", "265790267296391946810949632000000000", "127313963299399416749559771247411200000000000", "792786697595796795607377086400871488552960000000000000" ]	Superfactorials: product of first n factorials.
A000179	[ "1", "-1", "0", "1", "2", "13", "80", "579", "4738", "43387", "439792", "4890741", "59216642", "775596313", "10927434464", "164806435783", "2649391469058", "45226435601207", "817056406224416", "15574618910994665", "312400218671253762", "6577618644576902053", "145051250421230224304", "3343382818203784146955", "80399425364623070680706", "2013619745874493923699123" ]	Ménage numbers: a(0) = 1, a(1) = -1, and for n >= 2, a(n) = number of permutations s of [0, ..., n-1] such that s(i) != i and s(i) != i+1 (mod n) for all i.
A000180	[ "1", "2", "13", "116", "1393", "20894", "376093", "7897952", "189550849", "5117872922", "153536187661", "5066694192812", "182400990941233", "7113638646708086", "298772823161739613", "13444777042278282584", "645349298029357564033", "32912814199497235765682" ]	Expansion of E.g.f. exp(-x)/(1-3x).
A000181	[ "2", "15", "60", "469", "3660", "32958", "328920", "3614490", "43341822", "563144725", "7880897892", "118177520295", "1890389939000", "32130521850972", "578260307815920", "10985555094348948", "219687969344126490", "4613039009310624795", "101479234383619208204", "2333872309936442446905" ]	Coefficients of ménage hit polynomials.
A000182	[ "1", "2", "16", "272", "7936", "353792", "22368256", "1903757312", "209865342976", "29088885112832", "4951498053124096", "1015423886506852352", "246921480190207983616", "70251601603943959887872", "23119184187809597841473536", "8713962757125169296170811392", "3729407703720529571097509625856" ]	Tangent (or "Zag") numbers: e.g.f. tan(x), also (up to signs) e.g.f. tanh(x).
A000183	[ "0", "0", "0", "1", "2", "20", "144", "1265", "12072", "126565", "1445100", "17875140", "238282730", "3407118041", "52034548064", "845569542593", "14570246018686", "265397214435860", "5095853023109484", "102877234050493609", "2178674876680100744", "48296053720501168037", "1118480911876659396600" ]	Number of discordant permutations of length n.
A000184	[ "2", "22", "164", "1030", "5868", "31388", "160648", "795846", "3845020", "18211380", "84876152", "390331292", "1775032504", "7995075960", "35715205136", "158401506118", "698102372988", "3059470021316", "13341467466520", "57918065919924", "250419305769512", "1078769490401032", "4631680461623664", "19825379450255900", "84622558822506328", "360270317908904328", "1530148541536781488", "6484511936352543096", "27423786092731382000", "115756362341775227888" ]	Number of genus 0 rooted maps with 3 faces with n vertices.
A000185	[ "2", "24", "140", "1232", "11268", "115056", "1284360", "15596208", "204710454", "2888897032", "43625578836", "702025263328", "11993721979336", "216822550325472", "4135337882588880", "82986434235959712", "1747976804189353962", "38559791049947726328", "889047923669760546140" ]	Coefficients of ménage hit polynomials.
A000186	[ "1", "0", "0", "2", "24", "552", "21280", "1073760", "70299264", "5792853248", "587159944704", "71822743499520", "10435273503677440", "1776780700509416448", "350461958856515690496", "79284041282622163140608", "20392765404792755583221760", "5917934230798104348783083520", "1924427226324694427836833857536" ]	Number of 3 X n Latin rectangles in which the first row is in order.
A000187	[ "2", "30", "3522", "1066590", "604935042", "551609685150", "737740947722562", "1360427147514751710", "3308161927353377294082", "10256718523496425979562270", "39490468691102039103925777602", "184856411587530526077816051412830", "1033888847501229495999134528615701122" ]	Generalized Euler numbers, c(5,n).
A000188	[ "1", "1", "1", "2", "1", "1", "1", "2", "3", "1", "1", "2", "1", "1", "1", "4", "1", "3", "1", "2", "1", "1", "1", "2", "5", "1", "3", "2", "1", "1", "1", "4", "1", "1", "1", "6", "1", "1", "1", "2", "1", "1", "1", "2", "3", "1", "1", "4", "7", "5", "1", "2", "1", "3", "1", "2", "1", "1", "1", "2", "1", "1", "3", "8", "1", "1", "1", "2", "1", "1", "1", "6", "1", "1", "5", "2", "1", "1", "1", "4", "9", "1", "1", "2", "1", "1", "1", "2", "1", "3" ]	(1) Number of solutions to x^2 == 0 (mod n). (2) Also square root of largest square dividing n. (3) Also max_{ d divides n } gcd(d, n/d).
A000189	[ "1", "1", "1", "2", "1", "1", "1", "4", "3", "1", "1", "2", "1", "1", "1", "4", "1", "3", "1", "2", "1", "1", "1", "4", "5", "1", "9", "2", "1", "1", "1", "8", "1", "1", "1", "6", "1", "1", "1", "4", "1", "1", "1", "2", "3", "1", "1", "4", "7", "5", "1", "2", "1", "9", "1", "4", "1", "1", "1", "2", "1", "1", "3", "16", "1", "1", "1", "2", "1", "1", "1", "12", "1", "1", "5", "2", "1", "1", "1", "4", "9", "1", "1", "2", "1", "1", "1", "4", "1", "3" ]	Number of solutions to x^3 == 0 (mod n).
A000190	[ "1", "1", "1", "2", "1", "1", "1", "4", "3", "1", "1", "2", "1", "1", "1", "8", "1", "3", "1", "2", "1", "1", "1", "4", "5", "1", "9", "2", "1", "1", "1", "8", "1", "1", "1", "6", "1", "1", "1", "4", "1", "1", "1", "2", "3", "1", "1", "8", "7", "5", "1", "2", "1", "9", "1", "4", "1", "1", "1", "2", "1", "1", "3", "16", "1", "1", "1", "2", "1", "1", "1", "12", "1", "1", "5", "2", "1", "1", "1", "8", "27", "1", "1", "2", "1", "1", "1", "4", "1", "3" ]	Number of solutions to x^4 == 0 (mod n).
A000191	[ "2", "46", "3362", "515086", "135274562", "54276473326", "30884386347362", "23657073914466766", "23471059057478981762", "29279357851856595135406", "44855282210826271011257762", "82787899853638102222862479246", "181184428895772987376073015175362", "463938847087789978515380344866258286" ]	Generalized tangent numbers d(3, n).
A000192	[ "2", "46", "7970", "3487246", "2849229890", "3741386059246", "7205584123783010", "19133892392367261646", "67000387673723462963330", "299131045427247559446422446", "1658470810032820740402966226850", "11179247066648898992009055586869646", "90035623994788132387893239340761189570" ]	Generalized Euler numbers c(6,n).
A000193	[ "0", "1", "1", "1", "2", "2", "2", "2", "2", "2", "2", "2", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4" ]	Nearest integer to log n.
A000194	[ "0", "1", "1", "2", "2", "2", "2", "3", "3", "3", "3", "3", "3", "4", "4", "4", "4", "4", "4", "4", "4", "5", "5", "5", "5", "5", "5", "5", "5", "5", "5", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "7", "7", "7", "7", "7", "7", "7", "7", "7", "7", "7", "7", "7", "7", "8", "8", "8", "8", "8", "8", "8", "8", "8", "8", "8", "8", "8", "8", "8", "8", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "10", "10", "10", "10", "10", "10", "10", "10" ]	n appears 2n times, for n >= 1; also nearest integer to square root of n.
A000195	[ "0", "0", "1", "1", "1", "1", "1", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4" ]	a(n) = floor(log(n)).
A000196	[ "0", "1", "1", "1", "2", "2", "2", "2", "2", "3", "3", "3", "3", "3", "3", "3", "4", "4", "4", "4", "4", "4", "4", "4", "4", "5", "5", "5", "5", "5", "5", "5", "5", "5", "5", "5", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "7", "7", "7", "7", "7", "7", "7", "7", "7", "7", "7", "7", "7", "7", "7", "8", "8", "8", "8", "8", "8", "8", "8", "8", "8", "8", "8", "8", "8", "8", "8", "8", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "10", "10" ]	Integer part of square root of n. Or, number of positive squares <= n. Or, n appears 2n+1 times.
A000197	[ "1", "1", "2", "720", "620448401733239439360000" ]	a(n) = (n!)!.
A000198	[ "1", "1", "3", "3", "5", "9", "21", "21", "81", "81", "81", "243", "243", "441", "1215", "1701", "1701", "6561", "6561", "6561", "45927", "45927", "45927", "137781", "137781", "229635", "1594323", "1594323", "1594323", "4782969", "4782969", "7971615", "14348907", "33480783", "33480783", "129140163", "129140163", "129140163" ]	Largest order of automorphism group of a tournament with n nodes.
A000199	[ "1", "3", "3", "7", "6", "12", "13", "20", "21", "34", "36", "51", "58", "78", "89", "118", "131", "171", "197", "245", "279", "349", "398", "486", "557", "671", "767", "920", "1046", "1244", "1421", "1667", "1898", "2225", "2525", "2937", "3333", "3856", "4367", "5034", "5683", "6521", "7365", "8409", "9473", "10795", "12133", "13775", "15466" ]	Coefficient of q^(2n-1) in the series expansion of Ramanujan's mock theta function f(q).
A000200	[ "0", "0", "1", "0", "1", "1", "3", "3", "9", "15", "38", "73", "174", "380", "915", "2124", "5134", "12281", "30010", "73401", "181835", "452165", "1133252", "2851710", "7215262", "18326528", "46750268", "119687146", "307528889", "792716193", "2049703887", "5314775856", "13817638615", "36012395538" ]	Number of bicentered hydrocarbons with n atoms.

A000101

[ "3", "5", "11", "29", "97", "127", "541", "907", "1151", "1361", "9587", "15727", "19661", "31469", "156007", "360749", "370373", "492227", "1349651", "1357333", "2010881", "4652507", "17051887", "20831533", "47326913", "122164969", "189695893", "191913031", "387096383", "436273291", "1294268779" ]

Record gaps between primes (upper end) (compare A002386, which gives lower ends of these gaps).

A000102

[ "0", "0", "0", "0", "1", "2", "5", "12", "27", "59", "127", "269", "563", "1167", "2400", "4903", "9960", "20135", "40534", "81300", "162538", "324020", "644282", "1278152", "2530407", "5000178", "9863763", "19427976", "38211861", "75059535", "147263905", "288609341", "565047233", "1105229439", "2159947998" ]

a(n) = number of compositions of n in which the maximum part size is 4.

A000103

[ "0", "0", "1", "1", "2", "5", "12", "34", "130", "525", "2472", "12400", "65619", "357504", "1992985", "11284042", "64719885", "375126827", "2194439398", "12941995397", "76890024027", "459873914230", "2767364341936", "16747182732792" ]

Number of n-node triangulations of sphere in which every node has degree >= 4.

A000104

[ "1", "1", "1", "2", "5", "12", "35", "107", "363", "1248", "4460", "16094", "58937", "217117", "805475", "3001127", "11230003", "42161529", "158781106", "599563893", "2269506062", "8609442688", "32725637373", "124621833354", "475368834568", "1816103345752", "6948228104703", "26618671505989", "102102788362303" ]

Number of n-celled free polyominoes without holes.

A000105

[ "1", "1", "1", "2", "5", "12", "35", "108", "369", "1285", "4655", "17073", "63600", "238591", "901971", "3426576", "13079255", "50107909", "192622052", "742624232", "2870671950", "11123060678", "43191857688", "168047007728", "654999700403", "2557227044764", "9999088822075", "39153010938487", "153511100594603" ]

Number of free polyominoes (or square animals) with n cells.

A000106

[ "1", "2", "5", "12", "30", "74", "188", "478", "1235", "3214", "8450", "22370", "59676", "160140", "432237", "1172436", "3194870", "8741442", "24007045", "66154654", "182864692", "506909562", "1408854940", "3925075510", "10959698606", "30665337738", "85967279447", "241433975446", "679192039401", "1913681367936", "5399924120339" ]

2nd power of rooted tree enumerator; number of linear forests of 2 rooted trees.

A000107

[ "0", "1", "2", "5", "13", "35", "95", "262", "727", "2033", "5714", "16136", "45733", "130046", "370803", "1059838", "3035591", "8710736", "25036934", "72069134", "207727501", "599461094", "1731818878", "5008149658", "14496034714", "41993925955", "121747732406", "353221737526", "1025471857282", "2978995353959", "8658997820084" ]

Number of rooted trees with n nodes and a single labeled node; pointed rooted trees; vertebrates.

A000108

[ "1", "1", "2", "5", "14", "42", "132", "429", "1430", "4862", "16796", "58786", "208012", "742900", "2674440", "9694845", "35357670", "129644790", "477638700", "1767263190", "6564120420", "24466267020", "91482563640", "343059613650", "1289904147324", "4861946401452", "18367353072152", "69533550916004", "263747951750360", "1002242216651368", "3814986502092304" ]

Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!).

A000109

[ "1", "1", "1", "2", "5", "14", "50", "233", "1249", "7595", "49566", "339722", "2406841", "17490241", "129664753", "977526957", "7475907149", "57896349553", "453382272049", "3585853662949", "28615703421545" ]

Number of simplicial polyhedra with n vertices; simple planar graphs with n vertices and 3n-6 edges; maximal simple planar graphs with n vertices; planar triangulations with n vertices; triangulations of the sphere with n vertices; 3-connected cubic planar graphs on 2n-4 vertices.

A000110

[ "1", "1", "2", "5", "15", "52", "203", "877", "4140", "21147", "115975", "678570", "4213597", "27644437", "190899322", "1382958545", "10480142147", "82864869804", "682076806159", "5832742205057", "51724158235372", "474869816156751", "4506715738447323", "44152005855084346", "445958869294805289", "4638590332229999353", "49631246523618756274" ]

Bell or exponential numbers: number of ways to partition a set of n labeled elements.

A000111

[ "1", "1", "1", "2", "5", "16", "61", "272", "1385", "7936", "50521", "353792", "2702765", "22368256", "199360981", "1903757312", "19391512145", "209865342976", "2404879675441", "29088885112832", "370371188237525", "4951498053124096", "69348874393137901", "1015423886506852352", "15514534163557086905", "246921480190207983616", "4087072509293123892361" ]

Euler or up/down numbers: e.g.f. sec(x) + tan(x). Also for n >= 2, half the number of alternating permutations on n letters (A001250).

A000112

[ "1", "1", "2", "5", "16", "63", "318", "2045", "16999", "183231", "2567284", "46749427", "1104891746", "33823827452", "1338193159771", "68275077901156", "4483130665195087" ]

Number of partially ordered sets ("posets") with n unlabeled elements.

A000113

[ "1", "3", "4", "3", "6", "12", "8", "6", "4", "18", "12", "12", "14", "24", "24", "6", "18", "12", "20", "18", "32", "36", "24", "24", "30", "42", "12", "24", "30", "72", "32", "12", "48", "54", "48", "12", "38", "60", "56", "36", "42", "96", "44", "36", "24", "72", "48", "24", "56", "90", "72", "42", "54", "36", "72", "48", "80", "90", "60", "72", "62", "96", "32", "12", "84", "144", "68" ]

Number of transformation groups of order n.

A000114

[ "3", "4", "6", "12", "12", "24", "24", "36", "36", "60", "48", "84", "72", "96", "96", "144", "108", "180", "144", "192", "180", "264", "192", "300", "252", "324", "288", "420", "288", "480", "384", "480", "432", "576", "432", "684", "540", "672", "576", "840", "576", "924", "720", "864", "792", "1104", "768", "1176", "900", "1152", "1008", "1404", "972", "1440" ]

Number of cusps of principal congruence subgroup GAMMA^{hat}(n).

A000115

[ "1", "1", "2", "2", "3", "4", "5", "6", "7", "8", "10", "11", "13", "14", "16", "18", "20", "22", "24", "26", "29", "31", "34", "36", "39", "42", "45", "48", "51", "54", "58", "61", "65", "68", "72", "76", "80", "84", "88", "92", "97", "101", "106", "110", "115", "120", "125", "130", "135", "140", "146", "151", "157", "162", "168", "174", "180", "186", "192", "198", "205", "211", "218", "224", "231", "238" ]

Denumerants: Expansion of 1/((1-x)*(1-x^2)*(1-x^5)).

A000116

[ "1", "2", "4", "8", "20", "56", "180", "596", "2068", "7316", "26272", "95420", "349716", "1290872", "4794088", "17896832", "67110932", "252648992", "954444608", "3616828364", "13743921632", "52357746896", "199911300472", "764877836468", "2932031358484", "11258999739560", "43303843861744", "166799988689300" ]

Number of even sequences with period 2n (bisection of A000013).

A000117

[ "1", "2", "4", "8", "18", "44", "122", "362", "1162", "3914", "13648", "48734", "176906", "649532", "2405236", "8964800", "33588234", "126390032", "477353376", "1808676326", "6872485104", "26179922024", "99957747388", "382443112538", "1466024067850", "5629516646996", "21651955485304", "83400061453514" ]

Number of even sequences with period 2n (bisection of A000011).

A000118

[ "1", "8", "24", "32", "24", "48", "96", "64", "24", "104", "144", "96", "96", "112", "192", "192", "24", "144", "312", "160", "144", "256", "288", "192", "96", "248", "336", "320", "192", "240", "576", "256", "24", "384", "432", "384", "312", "304", "480", "448", "144", "336", "768", "352", "288", "624", "576", "384", "96", "456", "744", "576", "336", "432", "960", "576", "192" ]

Number of ways of writing n as a sum of 4 squares; also theta series of four-dimensional cubic lattice Z^4.

A000119

[ "1", "1", "1", "2", "1", "2", "2", "1", "3", "2", "2", "3", "1", "3", "3", "2", "4", "2", "3", "3", "1", "4", "3", "3", "5", "2", "4", "4", "2", "5", "3", "3", "4", "1", "4", "4", "3", "6", "3", "5", "5", "2", "6", "4", "4", "6", "2", "5", "5", "3", "6", "3", "4", "4", "1", "5", "4", "4", "7", "3", "6", "6", "3", "8", "5", "5", "7", "2", "6", "6", "4", "8", "4", "6", "6", "2", "7", "5", "5", "8", "3", "6", "6", "3", "7", "4", "4", "5", "1", "5", "5", "4", "8", "4", "7", "7", "3", "9", "6", "6", "9", "3", "8", "8", "5" ]

Number of representations of n as a sum of distinct Fibonacci numbers.

A000120

[ "0", "1", "1", "2", "1", "2", "2", "3", "1", "2", "2", "3", "2", "3", "3", "4", "1", "2", "2", "3", "2", "3", "3", "4", "2", "3", "3", "4", "3", "4", "4", "5", "1", "2", "2", "3", "2", "3", "3", "4", "2", "3", "3", "4", "3", "4", "4", "5", "2", "3", "3", "4", "3", "4", "4", "5", "3", "4", "4", "5", "4", "5", "5", "6", "1", "2", "2", "3", "2", "3", "3", "4", "2", "3", "3", "4", "3", "4", "4", "5", "2", "3", "3", "4", "3", "4", "4", "5", "3", "4", "4", "5", "4", "5", "5", "6", "2", "3", "3", "4", "3", "4", "4", "5", "3" ]

1's-counting sequence: number of 1's in binary expansion of n (or the binary weight of n).

A000121

[ "1", "2", "2", "3", "3", "3", "4", "3", "4", "5", "4", "5", "4", "4", "6", "5", "6", "6", "5", "6", "4", "5", "7", "6", "8", "7", "6", "8", "6", "7", "8", "6", "7", "5", "5", "8", "7", "9", "9", "8", "10", "7", "8", "10", "8", "10", "8", "7", "10", "8", "9", "9", "7", "8", "5", "6", "9", "8", "11", "10", "9", "12", "9", "11", "13", "10", "12", "9", "8", "12", "10", "12", "12", "10", "12", "8", "9", "12", "10", "13", "11", "9", "12", "9", "10", "11", "8", "9", "6", "6", "10", "9" ]

Number of representations of n as a sum of Fibonacci numbers (1 is allowed twice as a part).

A000122

[ "1", "2", "0", "0", "2", "0", "0", "0", "0", "2", "0", "0", "0", "0", "0", "0", "2", "0", "0", "0", "0", "0", "0", "0", "0", "2", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "2", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "2", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "2", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "2", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "2", "0", "0", "0", "0" ]

Expansion of Jacobi theta function theta_3(x) = Sum_{m =-oo..oo} x^(m^2) (number of integer solutions to k^2 = n).

A000123

[ "1", "2", "4", "6", "10", "14", "20", "26", "36", "46", "60", "74", "94", "114", "140", "166", "202", "238", "284", "330", "390", "450", "524", "598", "692", "786", "900", "1014", "1154", "1294", "1460", "1626", "1828", "2030", "2268", "2506", "2790", "3074", "3404", "3734", "4124", "4514", "4964", "5414", "5938", "6462", "7060", "7658", "8350", "9042", "9828" ]

Number of binary partitions: number of partitions of 2n into powers of 2.

A000124

[ "1", "2", "4", "7", "11", "16", "22", "29", "37", "46", "56", "67", "79", "92", "106", "121", "137", "154", "172", "191", "211", "232", "254", "277", "301", "326", "352", "379", "407", "436", "466", "497", "529", "562", "596", "631", "667", "704", "742", "781", "821", "862", "904", "947", "991", "1036", "1082", "1129", "1177", "1226", "1276", "1327", "1379" ]

Central polygonal numbers (the Lazy Caterer's sequence): n(n+1)/2 + 1; or, maximal number of pieces formed when slicing a pancake with n cuts.

A000125

[ "1", "2", "4", "8", "15", "26", "42", "64", "93", "130", "176", "232", "299", "378", "470", "576", "697", "834", "988", "1160", "1351", "1562", "1794", "2048", "2325", "2626", "2952", "3304", "3683", "4090", "4526", "4992", "5489", "6018", "6580", "7176", "7807", "8474", "9178", "9920", "10701", "11522", "12384", "13288", "14235", "15226" ]

Cake numbers: maximal number of pieces resulting from n planar cuts through a cube (or cake): C(n+1,3) + n + 1.

A000126

[ "1", "2", "4", "8", "15", "27", "47", "80", "134", "222", "365", "597", "973", "1582", "2568", "4164", "6747", "10927", "17691", "28636", "46346", "75002", "121369", "196393", "317785", "514202", "832012", "1346240", "2178279", "3524547", "5702855", "9227432", "14930318", "24157782", "39088133", "63245949" ]

A nonlinear binomial sum.

A000127

[ "1", "2", "4", "8", "16", "31", "57", "99", "163", "256", "386", "562", "794", "1093", "1471", "1941", "2517", "3214", "4048", "5036", "6196", "7547", "9109", "10903", "12951", "15276", "17902", "20854", "24158", "27841", "31931", "36457", "41449", "46938", "52956", "59536", "66712", "74519", "82993", "92171", "102091", "112792", "124314", "136698" ]

Maximal number of regions obtained by joining n points around a circle by straight lines. Also number of regions in 4-space formed by n-1 hyperplanes.

A000128

[ "1", "2", "4", "8", "16", "31", "58", "105", "185", "319", "541", "906", "1503", "2476", "4058", "6626", "10790", "17537", "28464", "46155", "74791", "121137", "196139", "317508", "513901", "831686", "1345888", "2177900", "3524140", "5702419", "9226966", "14929821", "24157253", "39087571", "63245353", "102333486" ]

A nonlinear binomial sum.

A000129

[ "0", "1", "2", "5", "12", "29", "70", "169", "408", "985", "2378", "5741", "13860", "33461", "80782", "195025", "470832", "1136689", "2744210", "6625109", "15994428", "38613965", "93222358", "225058681", "543339720", "1311738121", "3166815962", "7645370045", "18457556052", "44560482149", "107578520350", "259717522849" ]

Pell numbers: a(0) = 0, a(1) = 1; for n > 1, a(n) = 2*a(n-1) + a(n-2).

A000130

[ "0", "0", "1", "2", "5", "20", "115", "790", "6217", "55160", "545135", "5938490", "70686805", "912660508", "12702694075", "189579135710", "3019908731105", "51139445487680", "917345570926087", "17376071107513090", "346563420097249645", "7259714390232227300", "159352909727731210835", "3657569576966074846118" ]

One-half the number of permutations of length n with exactly 1 rising or falling successions.

A000131

[ "2", "5", "21", "61", "214", "669", "2240", "7330", "24695", "83257", "284928", "981079", "3410990", "11937328", "42075242", "149171958", "531866972", "1905842605", "6861162880", "24805692978", "90035940227", "327987890608", "1198853954688", "4395797189206", "16165195705544", "59609156824273", "220373268471398", "816677398144221" ]

Number of asymmetrical dissections of n-gon.

A000132

[ "1", "10", "40", "80", "90", "112", "240", "320", "200", "250", "560", "560", "400", "560", "800", "960", "730", "480", "1240", "1520", "752", "1120", "1840", "1600", "1200", "1210", "2000", "2240", "1600", "1680", "2720", "3200", "1480", "1440", "3680", "3040", "2250", "2800", "3280", "4160", "2800", "1920", "4320", "5040", "2800", "3472", "5920" ]

Number of ways of writing n as a sum of 5 squares.

A000133

[ "2", "5", "30", "2288", "67172352", "144115192303714304", "1329227995784915891206435945914040320", "226156424291633194186662080095093570364871077725232774230036394136943198208" ]

Number of Boolean functions of n variables.

A000134

[ "2", "6", "9", "12", "15", "18", "21", "24", "27", "31", "34", "37", "40", "43", "46", "49", "53", "56", "59", "62", "65", "68", "71", "75", "78", "81", "84", "87", "90", "93", "97", "100", "103", "106", "109", "112", "115", "119", "122", "125", "128", "131", "134", "137", "141", "144", "147", "150", "153", "156", "159", "163", "166", "169", "172", "175", "178", "181", "185", "188" ]

Positive zeros of Bessel function of order 0 rounded to nearest integer.

A000135

[ "1", "2", "6", "13", "24", "42", "73", "125", "204", "324", "511", "801", "1228", "1856", "2780", "4135", "6084", "8873", "12847", "18481", "26416", "37473", "52871", "74216", "103596", "143841", "198839", "273654", "374987", "511735", "695559", "941932", "1271139", "1709474", "2291195", "3061385", "4078152", "5416322" ]

Number of partitions into non-integral powers.

A000136

[ "1", "2", "6", "16", "50", "144", "462", "1392", "4536", "14060", "46310", "146376", "485914", "1557892", "5202690", "16861984", "56579196", "184940388", "622945970", "2050228360", "6927964218", "22930109884", "77692142980", "258360586368", "877395996200", "2929432171328", "9968202968958", "33396290888520", "113837957337750" ]

Number of ways of folding a strip of n labeled stamps.

A000137

[ "1", "2", "6", "18", "58", "186", "614", "2034", "6818", "22970", "77858", "264970", "905294", "3102434", "10661370", "36722642", "126752218", "438294018", "1518032598", "5265341314", "18286911130", "63586988434", "221342104842", "771235606050", "2689688538646", "9388096331642" ]

Series-parallel numbers.

A000138

[ "1", "1", "2", "6", "18", "90", "540", "3780", "31500", "283500", "2835000", "31185000", "372972600", "4848643800", "67881013200", "1018215198000", "16294848570000", "277012425690000", "4986223662420000", "94738249585980000", "1894745192712372000" ]

Expansion of e.g.f. exp(-x^4/4)/(1-x).

A000139

[ "2", "1", "2", "6", "22", "91", "408", "1938", "9614", "49335", "260130", "1402440", "7702632", "42975796", "243035536", "1390594458", "8038677054", "46892282815", "275750636070", "1633292229030", "9737153323590", "58392041019795", "352044769046880", "2132866978427640" ]

a(n) = 2*(3*n)! / ((2*n+1)!*(n+1)!).

A000140

[ "1", "1", "2", "6", "22", "101", "573", "3836", "29228", "250749", "2409581", "25598186", "296643390", "3727542188", "50626553988", "738680521142", "11501573822788", "190418421447330", "3344822488498265", "62119523114983224", "1214967840930909302", "24965661442811799655", "538134522243713149122" ]

Kendall-Mann numbers: the maximal number of inversions in a permutation on n letters is floor(n*(n-1)/4); a(n) is the number of permutations with this many inversions.

A000141

[ "1", "12", "60", "160", "252", "312", "544", "960", "1020", "876", "1560", "2400", "2080", "2040", "3264", "4160", "4092", "3480", "4380", "7200", "6552", "4608", "8160", "10560", "8224", "7812", "10200", "13120", "12480", "10104", "14144", "19200", "16380", "11520", "17400", "24960", "18396", "16440", "24480", "27200" ]

Number of ways of writing n as a sum of 6 squares.

A000142

[ "1", "1", "2", "6", "24", "120", "720", "5040", "40320", "362880", "3628800", "39916800", "479001600", "6227020800", "87178291200", "1307674368000", "20922789888000", "355687428096000", "6402373705728000", "121645100408832000", "2432902008176640000", "51090942171709440000", "1124000727777607680000" ]

Factorial numbers: n! = 1*2*3*4*...*n (order of symmetric group S_n, number of permutations of n letters).

A000143

[ "1", "16", "112", "448", "1136", "2016", "3136", "5504", "9328", "12112", "14112", "21312", "31808", "35168", "38528", "56448", "74864", "78624", "84784", "109760", "143136", "154112", "149184", "194688", "261184", "252016", "246176", "327040", "390784", "390240", "395136", "476672", "599152", "596736", "550368", "693504", "859952" ]

Number of ways of writing n as a sum of 8 squares.

A000144

[ "1", "20", "180", "960", "3380", "8424", "16320", "28800", "52020", "88660", "129064", "175680", "262080", "386920", "489600", "600960", "840500", "1137960", "1330420", "1563840", "2050344", "2611200", "2986560", "3358080", "4194240", "5318268", "5878440", "6299520", "7862400", "9619560" ]

Number of ways of writing n as a sum of 10 squares.

A000145

[ "1", "24", "264", "1760", "7944", "25872", "64416", "133056", "253704", "472760", "825264", "1297056", "1938336", "2963664", "4437312", "6091584", "8118024", "11368368", "15653352", "19822176", "24832944", "32826112", "42517728", "51425088", "61903776", "78146664", "98021616" ]

Number of ways of writing n as a sum of 12 squares.

A000146

[ "1", "1", "1", "1", "1", "1", "2", "-6", "56", "-528", "6193", "-86579", "1425518", "-27298230", "601580875", "-15116315766", "429614643062", "-13711655205087", "488332318973594", "-19296579341940067", "841693047573682616", "-40338071854059455412", "2115074863808199160561", "-120866265222965259346026" ]

From von Staudt-Clausen representation of Bernoulli numbers: a(n) = Bernoulli(2n) + Sum_{(p-1)|2n} 1/p.

A000147

[ "0", "0", "0", "0", "0", "1", "2", "7", "14", "32", "58", "110", "187", "322", "519", "839", "1302", "2015", "3032", "4542", "6668", "9738", "14006", "20036", "28324", "39830", "55473", "76875", "105692", "144629", "196585", "266038", "357952", "479664", "639519", "849425", "1123191", "1479972", "1942284", "2540674", "3311415" ]

Number of trees of diameter 5.

A000148

[ "1", "2", "7", "15", "28", "45", "70", "100", "138", "183", "242", "310", "388", "481", "583", "701", "838", "984", "1152", "1337", "1535", "1757", "2001", "2262", "2545", "2855", "3183", "3540", "3926", "4335", "4770", "5233", "5728", "6248", "6801", "7388", "8005", "8658", "9345", "10064", "10824", "11620", "12452", "13324", "14236" ]

Number of partitions into non-integral powers.

A000149

[ "1", "2", "7", "20", "54", "148", "403", "1096", "2980", "8103", "22026", "59874", "162754", "442413", "1202604", "3269017", "8886110", "24154952", "65659969", "178482300", "485165195", "1318815734", "3584912846", "9744803446", "26489122129", "72004899337", "195729609428", "532048240601", "1446257064291" ]

a(n) = floor(e^n).

A000150

[ "0", "0", "1", "2", "7", "20", "66", "212", "715", "2424", "8398", "29372", "104006", "371384", "1337220", "4847208", "17678835", "64821680", "238819350", "883629164", "3282060210", "12233125112", "45741281820", "171529777432", "644952073662", "2430973096720", "9183676536076" ]

Number of dissections of an n-gon, rooted at an exterior edge, asymmetric with respect to that edge.

A000151

[ "1", "2", "7", "26", "107", "458", "2058", "9498", "44947", "216598", "1059952", "5251806", "26297238", "132856766", "676398395", "3466799104", "17873508798", "92630098886", "482292684506", "2521610175006", "13233573019372", "69687684810980", "368114512431638", "1950037285256658", "10357028326495097", "55140508518522726", "294219119815868952", "1573132563600386854", "8427354035116949486", "45226421721391554194" ]

Number of oriented rooted trees with n nodes. Also rooted trees with n nodes and 2-colored non-root nodes.

A000152

[ "1", "32", "480", "4480", "29152", "140736", "525952", "1580800", "3994080", "8945824", "18626112", "36714624", "67978880", "118156480", "197120256", "321692928", "509145568", "772845120", "1143441760", "1681379200", "2428524096", "3392205824", "4658843520", "6411152640" ]

Number of ways of writing n as a sum of 16 squares.

A000153

[ "0", "1", "2", "7", "32", "181", "1214", "9403", "82508", "808393", "8743994", "103459471", "1328953592", "18414450877", "273749755382", "4345634192131", "73362643649444", "1312349454922513", "24796092486996338", "493435697986613143", "10315043624498196944" ]

a(n) = n*a(n-1) + (n-2)*a(n-2), with a(0) = 0, a(1) = 1.

A000154

[ "1", "1", "2", "7", "35", "228", "1834", "17382", "195866", "2487832", "35499576", "562356672", "9794156448", "186025364016", "3826961710272", "84775065603888", "2011929826983504" ]

Erroneous version of A003713.

A000155

[ "0", "1", "2", "7", "44", "361", "3654", "44207", "622552", "10005041", "180713290", "3624270839", "79914671748", "1921576392793", "50040900884366", "1403066801155039", "42142044935535536", "1349948504738292193", "45940391206037470098" ]

Nearest integer to modified Bessel function K_n(1).

A000156

[ "1", "48", "1104", "16192", "170064", "1362336", "8662720", "44981376", "195082320", "721175536", "2319457632", "6631997376", "17231109824", "41469483552", "93703589760", "200343312768", "407488018512", "793229226336", "1487286966928", "2697825744960", "4744779429216" ]

Number of ways of writing n as a sum of 24 squares.

A000157

[ "1", "2", "7", "111", "308063", "100126976263592", "131867858014413288241233435594064", "2804519150441879896741320496043335469582478995067528608051651559815168" ]

Number of Boolean functions of n variables.

A000158

[ "1", "2", "8", "19", "41", "78", "134", "218", "339", "506", "730", "1023", "1397", "1884", "2477", "3218", "4118", "5192", "6486", "8010", "9795", "11888", "14302", "17066", "20256", "23889", "27999", "32637", "37863", "43695", "50218", "57495", "65545", "74431", "84257", "95030", "106840", "119799", "133941", "149311", "166071" ]

Number of partitions into non-integral powers.

A000159

[ "2", "8", "20", "152", "994", "7888", "70152", "695760", "7603266", "90758872", "1174753372", "16386899368", "245046377410", "3910358788256", "66323124297872", "1191406991067168", "22596344660865282", "451208920617687720", "9461897733571886372", "207894669895136763704", "4776019866458134139042" ]

Coefficients of ménage hit polynomials.

A000160

[ "1", "2", "8", "21", "48", "99", "186", "330", "556", "895", "1397", "2107", "3097", "4459", "6264", "8644", "11760", "15742", "20790", "27128", "34993", "44664", "56473", "70784", "87995", "108564", "132970", "161828", "195686", "235274", "281349", "334682", "396202", "466849", "547712", "639935", "744716", "863443" ]

Number of partitions into non-integral powers.

A000161

[ "1", "1", "1", "0", "1", "1", "0", "0", "1", "1", "1", "0", "0", "1", "0", "0", "1", "1", "1", "0", "1", "0", "0", "0", "0", "2", "1", "0", "0", "1", "0", "0", "1", "0", "1", "0", "1", "1", "0", "0", "1", "1", "0", "0", "0", "1", "0", "0", "0", "1", "2", "0", "1", "1", "0", "0", "0", "0", "1", "0", "0", "1", "0", "0", "1", "2", "0", "0", "1", "0", "0", "0", "1", "1", "1", "0", "0", "0", "0", "0", "1", "1", "1", "0", "0", "2", "0", "0", "0", "1", "1", "0", "0", "0", "0", "0", "0", "1", "1", "0", "2", "1", "0", "0", "1", "0", "1", "0" ]

Number of partitions of n into 2 squares.

A000162

[ "1", "1", "2", "8", "29", "166", "1023", "6922", "48311", "346543", "2522522", "18598427", "138462649", "1039496297", "7859514470", "59795121480" ]

Number of 3-dimensional polyominoes (or polycubes) with n cells.

A000163

[ "2", "8", "34", "136", "538", "2080", "7970", "30224", "113874", "426520", "1590338", "5906640", "21866794", "80725280", "297292610", "1092539736", "4007553922", "14675748416", "53663197698", "195962433328", "714734393378", "2603994002280", "9477643739746", "34463730424720" ]

Series-parallel numbers.

A000164

[ "1", "1", "1", "1", "1", "1", "1", "0", "1", "2", "1", "1", "1", "1", "1", "0", "1", "2", "2", "1", "1", "1", "1", "0", "1", "2", "2", "2", "0", "2", "1", "0", "1", "2", "2", "1", "2", "1", "2", "0", "1", "3", "1", "1", "1", "2", "1", "0", "1", "2", "3", "2", "1", "2", "3", "0", "1", "2", "1", "2", "0", "2", "2", "0", "1", "3", "3", "1", "2", "2", "1", "0", "2", "2", "3", "2", "1", "2", "1", "0", "1", "4", "2", "2", "1", "2", "3", "0", "1", "4", "3", "1", "0", "1", "2", "0", "1", "2", "3", "3", "2", "4", "2", "0", "2" ]

Number of partitions of n into 3 squares (allowing part zero).

A000165

[ "1", "2", "8", "48", "384", "3840", "46080", "645120", "10321920", "185794560", "3715891200", "81749606400", "1961990553600", "51011754393600", "1428329123020800", "42849873690624000", "1371195958099968000", "46620662575398912000", "1678343852714360832000", "63777066403145711616000" ]

Double factorial of even numbers: (2n)!! = 2^n*n!.

A000166

[ "1", "0", "1", "2", "9", "44", "265", "1854", "14833", "133496", "1334961", "14684570", "176214841", "2290792932", "32071101049", "481066515734", "7697064251745", "130850092279664", "2355301661033953", "44750731559645106", "895014631192902121", "18795307255050944540", "413496759611120779881", "9510425471055777937262" ]

Subfactorial or rencontres numbers, or derangements: number of permutations of n elements with no fixed points.

A000167

[ "0", "0", "0", "1", "2", "9", "49", "306", "2188", "17810", "162482", "1642635", "18231462", "220420179", "2883693795", "40592133316", "611765693528", "9828843229764", "167702100599524", "3028466654021205", "57708568527002410", "1157199837194069405" ]

Nearest integer to modified Bessel function K_n(2).

A000168

[ "1", "2", "9", "54", "378", "2916", "24057", "208494", "1876446", "17399772", "165297834", "1602117468", "15792300756", "157923007560", "1598970451545", "16365932856990", "169114639522230", "1762352559231660", "18504701871932430", "195621134074714260", "2080697516976506220", "22254416920705240440", "239234981897581334730", "2583737804493878415084" ]

a(n) = 2*3^n*(2*n)!/(n!*(n+2)!).

A000169

[ "1", "2", "9", "64", "625", "7776", "117649", "2097152", "43046721", "1000000000", "25937424601", "743008370688", "23298085122481", "793714773254144", "29192926025390625", "1152921504606846976", "48661191875666868481", "2185911559738696531968", "104127350297911241532841", "5242880000000000000000000" ]

Number of labeled rooted trees with n nodes: n^(n-1).

A000170

[ "1", "1", "0", "0", "2", "10", "4", "40", "92", "352", "724", "2680", "14200", "73712", "365596", "2279184", "14772512", "95815104", "666090624", "4968057848", "39029188884", "314666222712", "2691008701644", "24233937684440", "227514171973736", "2207893435808352", "22317699616364044", "234907967154122528" ]

Number of ways of placing n nonattacking queens on an n X n board.

A000171

[ "1", "0", "0", "1", "2", "0", "0", "10", "36", "0", "0", "720", "5600", "0", "0", "703760", "11220000", "0", "0", "9168331776", "293293716992", "0", "0", "1601371799340544", "102484848265030656", "0", "0", "3837878966366932639744", "491247277315343649710080", "0", "0" ]

Number of self-complementary graphs with n nodes.

A000172

[ "1", "2", "10", "56", "346", "2252", "15184", "104960", "739162", "5280932", "38165260", "278415920", "2046924400", "15148345760", "112738423360", "843126957056", "6332299624282", "47737325577620", "361077477684436", "2739270870994736", "20836827035351596" ]

Franel number a(n) = Sum_{k = 0..n} binomial(n,k)^3.

A000173

[ "30", "1482", "2418", "24180", "35238", "263820", "395730", "473298", "698130", "763620", "2212026", "2233554", "172459210", "209524210", "341354790", "384121920", "525150234", "530946330", "582129630", "1799281330", "2069510520", "2514290520", "3344596854", "5251502340" ]

Unitary-sociable numbers (smallest member of each cycle).

A000174

[ "1", "1", "1", "1", "2", "2", "1", "1", "2", "2", "2", "2", "2", "3", "2", "1", "3", "3", "3", "3", "4", "3", "3", "2", "2", "4", "4", "4", "4", "5", "3", "3", "4", "3", "5", "5", "5", "6", "5", "3", "5", "5", "5", "6", "5", "6", "5", "4", "4", "6", "7", "6", "8", "8", "7", "5", "6", "5", "8", "8", "4", "9", "7", "6", "7", "7", "8", "9", "9", "7", "9", "6", "7", "8", "10", "9", "9", "11", "7", "8", "8", "8", "11", "11", "9", "12", "11", "7", "9", "9", "12", "13", "9", "10", "11", "8", "6", "10", "13", "13", "13" ]

Number of partitions of n into 5 squares.

A000175

[ "1", "1", "2", "11", "38", "946", "4580", "202738", "3786092", "261868876", "1992367192", "2381255244240", "21411255538848", "2902625722978656", "451716954504285504", "319933105641374465472", "3761845343198709705600" ]

Related to zeros of Bessel function.

A000176

[ "2", "11", "46", "128", "272", "522", "904", "1408", "2160", "3154", "4306", "5888", "7888", "10012", "12888", "16384", "19680", "24354", "29866", "34816", "41888", "49778", "56744", "66816", "78000", "87358", "100602", "115712", "128112", "145804", "165712", "180224", "203040", "228964", "246932", "276480" ]

Generalized tangent numbers d_(n,2).

A000177

[ "1", "1", "1", "1", "2", "2", "2", "1", "2", "3", "2", "2", "3", "3", "3", "2", "3", "4", "4", "3", "5", "5", "4", "3", "4", "5", "5", "5", "5", "7", "6", "4", "5", "6", "6", "6", "9", "7", "8", "6", "6", "9", "7", "7", "9", "10", "8", "7", "7", "9", "10", "9", "11", "12", "13", "8", "10", "11", "10", "11", "11", "12", "13", "11", "9", "14", "13", "12", "16", "14", "14", "12", "13", "13", "15", "15", "15", "18", "16", "13", "14", "18", "15", "17", "20", "17", "21", "15", "15" ]

Number of partitions of n into 6 squares.

A000178

[ "1", "1", "2", "12", "288", "34560", "24883200", "125411328000", "5056584744960000", "1834933472251084800000", "6658606584104736522240000000", "265790267296391946810949632000000000", "127313963299399416749559771247411200000000000", "792786697595796795607377086400871488552960000000000000" ]

Superfactorials: product of first n factorials.

A000179

[ "1", "-1", "0", "1", "2", "13", "80", "579", "4738", "43387", "439792", "4890741", "59216642", "775596313", "10927434464", "164806435783", "2649391469058", "45226435601207", "817056406224416", "15574618910994665", "312400218671253762", "6577618644576902053", "145051250421230224304", "3343382818203784146955", "80399425364623070680706", "2013619745874493923699123" ]

Ménage numbers: a(0) = 1, a(1) = -1, and for n >= 2, a(n) = number of permutations s of [0, ..., n-1] such that s(i) != i and s(i) != i+1 (mod n) for all i.

A000180

[ "1", "2", "13", "116", "1393", "20894", "376093", "7897952", "189550849", "5117872922", "153536187661", "5066694192812", "182400990941233", "7113638646708086", "298772823161739613", "13444777042278282584", "645349298029357564033", "32912814199497235765682" ]

Expansion of E.g.f. exp(-x)/(1-3x).

A000181

[ "2", "15", "60", "469", "3660", "32958", "328920", "3614490", "43341822", "563144725", "7880897892", "118177520295", "1890389939000", "32130521850972", "578260307815920", "10985555094348948", "219687969344126490", "4613039009310624795", "101479234383619208204", "2333872309936442446905" ]

Coefficients of ménage hit polynomials.

A000182

[ "1", "2", "16", "272", "7936", "353792", "22368256", "1903757312", "209865342976", "29088885112832", "4951498053124096", "1015423886506852352", "246921480190207983616", "70251601603943959887872", "23119184187809597841473536", "8713962757125169296170811392", "3729407703720529571097509625856" ]

Tangent (or "Zag") numbers: e.g.f. tan(x), also (up to signs) e.g.f. tanh(x).

A000183

[ "0", "0", "0", "1", "2", "20", "144", "1265", "12072", "126565", "1445100", "17875140", "238282730", "3407118041", "52034548064", "845569542593", "14570246018686", "265397214435860", "5095853023109484", "102877234050493609", "2178674876680100744", "48296053720501168037", "1118480911876659396600" ]

Number of discordant permutations of length n.

A000184

[ "2", "22", "164", "1030", "5868", "31388", "160648", "795846", "3845020", "18211380", "84876152", "390331292", "1775032504", "7995075960", "35715205136", "158401506118", "698102372988", "3059470021316", "13341467466520", "57918065919924", "250419305769512", "1078769490401032", "4631680461623664", "19825379450255900", "84622558822506328", "360270317908904328", "1530148541536781488", "6484511936352543096", "27423786092731382000", "115756362341775227888" ]

Number of genus 0 rooted maps with 3 faces with n vertices.

A000185

[ "2", "24", "140", "1232", "11268", "115056", "1284360", "15596208", "204710454", "2888897032", "43625578836", "702025263328", "11993721979336", "216822550325472", "4135337882588880", "82986434235959712", "1747976804189353962", "38559791049947726328", "889047923669760546140" ]

Coefficients of ménage hit polynomials.

A000186

[ "1", "0", "0", "2", "24", "552", "21280", "1073760", "70299264", "5792853248", "587159944704", "71822743499520", "10435273503677440", "1776780700509416448", "350461958856515690496", "79284041282622163140608", "20392765404792755583221760", "5917934230798104348783083520", "1924427226324694427836833857536" ]

Number of 3 X n Latin rectangles in which the first row is in order.

A000187

[ "2", "30", "3522", "1066590", "604935042", "551609685150", "737740947722562", "1360427147514751710", "3308161927353377294082", "10256718523496425979562270", "39490468691102039103925777602", "184856411587530526077816051412830", "1033888847501229495999134528615701122" ]

Generalized Euler numbers, c(5,n).

A000188

[ "1", "1", "1", "2", "1", "1", "1", "2", "3", "1", "1", "2", "1", "1", "1", "4", "1", "3", "1", "2", "1", "1", "1", "2", "5", "1", "3", "2", "1", "1", "1", "4", "1", "1", "1", "6", "1", "1", "1", "2", "1", "1", "1", "2", "3", "1", "1", "4", "7", "5", "1", "2", "1", "3", "1", "2", "1", "1", "1", "2", "1", "1", "3", "8", "1", "1", "1", "2", "1", "1", "1", "6", "1", "1", "5", "2", "1", "1", "1", "4", "9", "1", "1", "2", "1", "1", "1", "2", "1", "3" ]

(1) Number of solutions to x^2 == 0 (mod n). (2) Also square root of largest square dividing n. (3) Also max_{ d divides n } gcd(d, n/d).

A000189

[ "1", "1", "1", "2", "1", "1", "1", "4", "3", "1", "1", "2", "1", "1", "1", "4", "1", "3", "1", "2", "1", "1", "1", "4", "5", "1", "9", "2", "1", "1", "1", "8", "1", "1", "1", "6", "1", "1", "1", "4", "1", "1", "1", "2", "3", "1", "1", "4", "7", "5", "1", "2", "1", "9", "1", "4", "1", "1", "1", "2", "1", "1", "3", "16", "1", "1", "1", "2", "1", "1", "1", "12", "1", "1", "5", "2", "1", "1", "1", "4", "9", "1", "1", "2", "1", "1", "1", "4", "1", "3" ]

Number of solutions to x^3 == 0 (mod n).

A000190

[ "1", "1", "1", "2", "1", "1", "1", "4", "3", "1", "1", "2", "1", "1", "1", "8", "1", "3", "1", "2", "1", "1", "1", "4", "5", "1", "9", "2", "1", "1", "1", "8", "1", "1", "1", "6", "1", "1", "1", "4", "1", "1", "1", "2", "3", "1", "1", "8", "7", "5", "1", "2", "1", "9", "1", "4", "1", "1", "1", "2", "1", "1", "3", "16", "1", "1", "1", "2", "1", "1", "1", "12", "1", "1", "5", "2", "1", "1", "1", "8", "27", "1", "1", "2", "1", "1", "1", "4", "1", "3" ]

Number of solutions to x^4 == 0 (mod n).

A000191

[ "2", "46", "3362", "515086", "135274562", "54276473326", "30884386347362", "23657073914466766", "23471059057478981762", "29279357851856595135406", "44855282210826271011257762", "82787899853638102222862479246", "181184428895772987376073015175362", "463938847087789978515380344866258286" ]

Generalized tangent numbers d(3, n).

A000192

[ "2", "46", "7970", "3487246", "2849229890", "3741386059246", "7205584123783010", "19133892392367261646", "67000387673723462963330", "299131045427247559446422446", "1658470810032820740402966226850", "11179247066648898992009055586869646", "90035623994788132387893239340761189570" ]

Generalized Euler numbers c(6,n).

A000193

[ "0", "1", "1", "1", "2", "2", "2", "2", "2", "2", "2", "2", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4" ]

Nearest integer to log n.

A000194

[ "0", "1", "1", "2", "2", "2", "2", "3", "3", "3", "3", "3", "3", "4", "4", "4", "4", "4", "4", "4", "4", "5", "5", "5", "5", "5", "5", "5", "5", "5", "5", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "7", "7", "7", "7", "7", "7", "7", "7", "7", "7", "7", "7", "7", "7", "8", "8", "8", "8", "8", "8", "8", "8", "8", "8", "8", "8", "8", "8", "8", "8", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "10", "10", "10", "10", "10", "10", "10", "10" ]

n appears 2n times, for n >= 1; also nearest integer to square root of n.

A000195

[ "0", "0", "1", "1", "1", "1", "1", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4" ]

a(n) = floor(log(n)).

A000196

[ "0", "1", "1", "1", "2", "2", "2", "2", "2", "3", "3", "3", "3", "3", "3", "3", "4", "4", "4", "4", "4", "4", "4", "4", "4", "5", "5", "5", "5", "5", "5", "5", "5", "5", "5", "5", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "7", "7", "7", "7", "7", "7", "7", "7", "7", "7", "7", "7", "7", "7", "7", "8", "8", "8", "8", "8", "8", "8", "8", "8", "8", "8", "8", "8", "8", "8", "8", "8", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "10", "10" ]

Integer part of square root of n. Or, number of positive squares <= n. Or, n appears 2n+1 times.

A000197

[ "1", "1", "2", "720", "620448401733239439360000" ]

a(n) = (n!)!.

A000198

[ "1", "1", "3", "3", "5", "9", "21", "21", "81", "81", "81", "243", "243", "441", "1215", "1701", "1701", "6561", "6561", "6561", "45927", "45927", "45927", "137781", "137781", "229635", "1594323", "1594323", "1594323", "4782969", "4782969", "7971615", "14348907", "33480783", "33480783", "129140163", "129140163", "129140163" ]

Largest order of automorphism group of a tournament with n nodes.

A000199

[ "1", "3", "3", "7", "6", "12", "13", "20", "21", "34", "36", "51", "58", "78", "89", "118", "131", "171", "197", "245", "279", "349", "398", "486", "557", "671", "767", "920", "1046", "1244", "1421", "1667", "1898", "2225", "2525", "2937", "3333", "3856", "4367", "5034", "5683", "6521", "7365", "8409", "9473", "10795", "12133", "13775", "15466" ]

Coefficient of q^(2n-1) in the series expansion of Ramanujan's mock theta function f(q).

A000200

[ "0", "0", "1", "0", "1", "1", "3", "3", "9", "15", "38", "73", "174", "380", "915", "2124", "5134", "12281", "30010", "73401", "181835", "452165", "1133252", "2851710", "7215262", "18326528", "46750268", "119687146", "307528889", "792716193", "2049703887", "5314775856", "13817638615", "36012395538" ]

Number of bicentered hydrocarbons with n atoms.