christopher/oeis · Datasets at Hugging Face

a-number stringlengths 7 7	sequence sequencelengths 1 377	description stringlengths 3 852
A000301	[ "1", "2", "2", "4", "8", "32", "256", "8192", "2097152", "17179869184", "36028797018963968", "618970019642690137449562112", "22300745198530623141535718272648361505980416", "13803492693581127574869511724554050904902217944340773110325048447598592" ]	a(n) = a(n-1)*a(n-2) with a(0) = 1, a(1) = 2; also a(n) = 2^Fibonacci(n).
A000302	[ "1", "4", "16", "64", "256", "1024", "4096", "16384", "65536", "262144", "1048576", "4194304", "16777216", "67108864", "268435456", "1073741824", "4294967296", "17179869184", "68719476736", "274877906944", "1099511627776", "4398046511104", "17592186044416", "70368744177664", "281474976710656" ]	Powers of 4: a(n) = 4^n.
A000303	[ "0", "1", "4", "16", "69", "348", "2016", "13357", "99376", "822040", "7477161", "74207208", "797771520", "9236662345", "114579019468", "1516103040832", "21314681315997", "317288088082404", "4985505271920096", "82459612672301845", "1432064398910663704", "26054771465540507272" ]	Number of permutations of [n] in which the longest increasing run has length 2.
A000304	[ "2", "3", "6", "18", "108", "1944", "209952", "408146688", "85691213438976", "34974584955819144511488", "2997014624388697307377363936018956288", "104819342594514896999066634490728502944926883876041385836544" ]	a(n) = a(n-1)*a(n-2).
A000305	[ "1", "4", "18", "89", "466", "2537", "14209", "81316", "473338", "2793454", "16674417", "100487896", "610549829", "3735850007", "23000055178", "142370597601", "885521350882", "5531501612071", "34686798239678", "218273864005214", "1377897874711437" ]	Number of certain rooted planar maps.
A000306	[ "1", "4", "19", "66", "219", "645", "1813", "4802", "12265", "30198", "72396", "169231", "387707", "871989", "1930868", "4215615", "9091410", "19389327", "40944999", "85691893", "177898521" ]	Number of trees of diameter 8.
A000307	[ "1", "1", "4", "22", "154", "1304", "12915", "146115", "1855570", "26097835", "402215465", "6734414075", "121629173423", "2355470737637", "48664218965021", "1067895971109199", "24795678053493443", "607144847919796830", "15630954703539323090", "421990078975569031642", "11918095123121138408128" ]	Number of 4-level labeled rooted trees with n leaves.
A000308	[ "1", "2", "3", "6", "36", "648", "139968", "3265173504", "296148833645101056", "135345882205792807436868315512832", "130876399105969522361889021452224949874232743897657526714368" ]	a(n) = a(n-1)a(n-2)a(n-3) with a(1)=1, a(2)=2 and a(3)=3.
A000309	[ "1", "1", "4", "24", "176", "1456", "13056", "124032", "1230592", "12629760", "133186560", "1436098560", "15774990336", "176028860416", "1990947110912", "22783499599872", "263411369705472", "3073132646563840", "36143187370967040", "428157758086840320", "5105072641718353920", "61228492804372561920" ]	Number of rooted planar bridgeless cubic maps with 2n nodes.
A000310	[ "1", "4", "26", "234", "2696", "37919", "630521", "12111114", "264051201", "6445170229", "174183891471", "5164718385337", "166737090160871", "5822980248613990", "218756388226681557", "8797723991458469015", "377159237609540937788", "17170729962232112834302", "827382365085791968518198", "42070004707327023844695198" ]	Coefficients of iterated exponentials.
A000311	[ "0", "1", "1", "4", "26", "236", "2752", "39208", "660032", "12818912", "282137824", "6939897856", "188666182784", "5617349020544", "181790703209728", "6353726042486272", "238513970965257728", "9571020586419012608", "408837905660444010496", "18522305410364986906624" ]	Schroeder's fourth problem; also series-reduced rooted trees with n labeled leaves; also number of total partitions of n.
A000312	[ "1", "1", "4", "27", "256", "3125", "46656", "823543", "16777216", "387420489", "10000000000", "285311670611", "8916100448256", "302875106592253", "11112006825558016", "437893890380859375", "18446744073709551616", "827240261886336764177", "39346408075296537575424", "1978419655660313589123979" ]	a(n) = n^n; number of labeled mappings from n points to themselves (endofunctions).
A000313	[ "0", "0", "0", "1", "4", "30", "220", "1855", "17304", "177996", "2002440", "24474285", "323060540", "4581585866", "69487385604", "1122488536715", "19242660629360", "348933579412440", "6673354706262864", "134252194678935321", "2834212998777523380", "62651024183503148470", "1447238658638922729580" ]	Number of permutations of length n with 3 consecutive ascending pairs.
A000314	[ "1", "1", "1", "4", "31", "362", "5676", "111982", "2666392", "74433564", "2384579440", "86248530296", "3476794472064", "154579941792256", "7514932528712896", "396595845237540600", "22581060079942183936", "1379771773100463174608", "90059660791562688208128", "6253914166368448348512064" ]	Number of mixed Husimi trees with n nodes; or labeled polygonal cacti with bridges.
A000315	[ "1", "1", "1", "4", "56", "9408", "16942080", "535281401856", "377597570964258816", "7580721483160132811489280", "5363937773277371298119673540771840" ]	Number of reduced Latin squares of order n; also number of labeled loops (quasigroups with an identity element) with a fixed identity element.
A000316	[ "1", "0", "4", "80", "4752", "440192", "59245120", "10930514688", "2649865335040", "817154768973824", "312426715251262464", "145060238642780180480", "80403174342119992692736", "52443098500204184915312640", "39764049487996490505336537088" ]	Two decks each have n kinds of cards, 2 of each kind. The first deck is laid out in order. The second deck is shuffled and laid out next to the first. A match occurs if a card from the second deck is next to a card of the same kind from the first deck. a(n) is the number of ways of achieving no matches.
A000317	[ "1", "2", "3", "7", "37", "1159", "1301767", "1693089917617", "2866551265129451657751739", "8217116155610406522540626640615749228405055996847" ]	a(n+1) = a(n)^2 - a(n) a(n-1) + a(n-1)^2.
A000318	[ "4", "128", "16384", "4456448", "2080374784", "1483911200768", "1501108249821184", "2044143848640217088", "3605459138582973251584", "7995891855149741436305408", "21776918737280678860353961984", "71454103701490016776039304265728", "278008871543597996197497752082448384" ]	Generalized tangent numbers d(4,n).
A000319	[ "1", "1", "74", "-1", "-2", "-3", "0", "1", "30", "-2", "-2", "29", "1", "4", "-6", "0", "1", "2", "-1", "-1", "-1", "-1", "-2", "-9", "0", "0", "1", "2", "-2", "-35", "-1", "-1", "-1", "-1", "-1", "-1", "-1", "-2", "-3", "0", "0", "1", "5", "-2", "-2", "3", "1", "1", "-4", "-1", "-1", "-1", "-1", "-1", "-1", "-1", "-1", "-2", "-3", "1", "2", "-1", "-1", "-1", "-1", "-1", "-1", "-1", "-1", "-1", "-1", "-1", "-1", "-1", "-1", "-1", "-1", "-1", "-1", "-2", "-3", "0", "1", "2", "-1", "-2", "-21", "-7", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0" ]	a(n) = floor(b(n)), where b(n) = tan(b(n-1)), b(0)=1.
A000320	[ "4", "272", "55744", "23750912", "17328937984", "19313964388352", "30527905292468224", "64955605537174126592", "179013508069217017790464", "620314831396713435870789632", "2639743384489464189324523208704", "13533573366345611477262311433961472", "82274260343572247169162187576069586944" ]	Generalized tangent numbers d(5,n).
A000321	[ "1", "-1", "-1", "5", "1", "-41", "31", "461", "-895", "-6481", "22591", "107029", "-604031", "-1964665", "17669471", "37341149", "-567425279", "-627491489", "19919950975", "2669742629", "-759627879679", "652838174519", "31251532771999", "-59976412450835", "-1377594095061119", "4256461892701199", "64623242860354751" ]	H_n(-1/2), where H_n(x) is Hermite polynomial of degree n.
A000322	[ "1", "1", "1", "1", "1", "5", "9", "17", "33", "65", "129", "253", "497", "977", "1921", "3777", "7425", "14597", "28697", "56417", "110913", "218049", "428673", "842749", "1656801", "3257185", "6403457", "12588865", "24749057", "48655365", "95653929", "188050673", "369697889", "726806913", "1428864769" ]	Pentanacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) + a(n-5) with a(0) = a(1) = a(2) = a(3) = a(4) = 1.
A000323	[ "5", "9", "21", "37", "69", "69", "89", "137", "177", "421", "481", "657", "749", "885", "1085", "1305", "1353", "1489", "1861", "2617", "2693", "3125", "5249", "5761", "7129", "8109", "9465", "9465", "10717", "12401", "12401", "16237", "16237", "24833", "30725", "35237", "46701", "47441", "47441", "61493", "67797", "67805", "67805" ]	Let A(n) = #{(i,j): i^2 + j^2 <= n}, V(n) = Pi*n, P(n) = A(n) - V(n); A000099 gives values of n where \|P(n)\| sets a new record; sequence gives A(A000099(n)).
A000324	[ "1", "5", "9", "49", "2209", "4870849", "23725150497409", "562882766124611619513723649", "316837008400094222150776738483768236006420971486980609" ]	A nonlinear recurrence: a(0) = 1, a(1) = 5, a(n) = a(n-1)^2 - 4*a(n-1) + 4 for n>1.
A000325	[ "1", "1", "2", "5", "12", "27", "58", "121", "248", "503", "1014", "2037", "4084", "8179", "16370", "32753", "65520", "131055", "262126", "524269", "1048556", "2097131", "4194282", "8388585", "16777192", "33554407", "67108838", "134217701", "268435428", "536870883", "1073741794", "2147483617" ]	a(n) = 2^n - n.
A000326	[ "0", "1", "5", "12", "22", "35", "51", "70", "92", "117", "145", "176", "210", "247", "287", "330", "376", "425", "477", "532", "590", "651", "715", "782", "852", "925", "1001", "1080", "1162", "1247", "1335", "1426", "1520", "1617", "1717", "1820", "1926", "2035", "2147", "2262", "2380", "2501", "2625", "2752", "2882", "3015", "3151" ]	Pentagonal numbers: a(n) = n(3n-1)/2.
A000327	[ "1", "5", "12", "23", "39", "62", "91", "127", "171", "228", "294", "370", "461", "561", "677", "811", "955", "1121", "1303", "1499", "1719", "1960", "2218", "2499", "2806", "3131", "3485", "3868", "4274", "4706", "5166", "5658", "6175", "6725", "7309", "7923", "8572", "9256", "9972", "10728", "11521", "12349", "13218", "14126", "15072" ]	Number of partitions into non-integral powers.
A000328	[ "1", "5", "13", "29", "49", "81", "113", "149", "197", "253", "317", "377", "441", "529", "613", "709", "797", "901", "1009", "1129", "1257", "1373", "1517", "1653", "1793", "1961", "2121", "2289", "2453", "2629", "2821", "3001", "3209", "3409", "3625", "3853", "4053", "4293", "4513", "4777", "5025", "5261", "5525", "5789", "6077", "6361", "6625" ]	Number of points of norm <= n^2 in square lattice.
A000329	[ "1", "2", "75", "-1", "-1", "-2", "1", "2", "31", "-1", "-2", "29", "1", "5", "-6", "1", "1", "3", "-1", "-1", "-1", "-1", "-1", "-9", "1", "1", "1", "2", "-2", "-35", "0", "0", "-1", "-1", "-1", "-1", "-1", "-1", "-2", "1", "1", "1", "5", "-1", "-2", "4", "1", "2", "-4", "0", "0", "0", "-1", "-1", "-1", "-1", "-1", "-1", "-2", "1", "3", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "-1" ]	Nearest integer to b(n), where b(n) = tan(b(n-1)), b(0) = 1.
A000330	[ "0", "1", "5", "14", "30", "55", "91", "140", "204", "285", "385", "506", "650", "819", "1015", "1240", "1496", "1785", "2109", "2470", "2870", "3311", "3795", "4324", "4900", "5525", "6201", "6930", "7714", "8555", "9455", "10416", "11440", "12529", "13685", "14910", "16206", "17575", "19019", "20540", "22140", "23821", "25585", "27434", "29370" ]	Square pyramidal numbers: a(n) = 0^2 + 1^2 + 2^2 + ... + n^2 = n(n+1)(2*n+1)/6.
A000331	[ "5", "14", "1026", "4324", "311387", "6425694", "579783114", "4028104212", "7315072725560", "61358264615344", "9569450876916944", "1632353370882506848", "1365475358484643531856", "15211641461623992544160", "74766806258361827981250240", "936580261005146914634459520", "6083678228249789825160175706880", "1936651082361926268672618636234240", "688115696843061332335070140230720000", "10517068622936239459488783307672335360", "2913914903970372007778735454555848514846720" ]	Related to zeros of Bessel function.
A000332	[ "0", "0", "0", "0", "1", "5", "15", "35", "70", "126", "210", "330", "495", "715", "1001", "1365", "1820", "2380", "3060", "3876", "4845", "5985", "7315", "8855", "10626", "12650", "14950", "17550", "20475", "23751", "27405", "31465", "35960", "40920", "46376", "52360", "58905", "66045", "73815", "82251", "91390", "101270", "111930", "123410" ]	Binomial coefficient binomial(n,4) = n(n-1)(n-2)*(n-3)/24.
A000333	[ "1", "5", "15", "40", "98", "237", "534", "1185", "2554", "5391", "11117", "22556", "44858", "88000", "170107", "324547", "611755", "1140382", "2103554", "3842826", "6955918", "12483075", "22220002", "39248230", "68819781", "119839422", "207304370", "356356801", "608901907", "1034452712", "1747764522", "2937370605", "4911675955", "8173032301" ]	Number of partitions into non-integral powers.
A000334	[ "1", "5", "15", "45", "120", "326", "835", "2145", "5345", "13220", "32068", "76965", "181975", "425490", "982615", "2245444", "5077090", "11371250", "25235790", "55536870", "121250185", "262769080", "565502405", "1209096875", "2569270050", "5427963902", "11404408525", "23836421895", "49573316740", "102610460240" ]	Number of 4-dimensional partitions of n.
A000335	[ "1", "5", "15", "45", "120", "331", "855", "2214", "5545", "13741", "33362", "80091", "189339", "442799", "1023192", "2340904", "5302061", "11902618", "26488454", "58479965", "128120214", "278680698", "602009786", "1292027222", "2755684669", "5842618668", "12317175320", "25825429276", "53865355154", "111786084504", "230867856903", "474585792077", "971209629993" ]	Euler transform of A000292.
A000336	[ "1", "2", "3", "4", "24", "576", "165888", "9172942848", "21035720123168587776", "18437563379178327736384102280592359424", "590180110002114158896983994712576414865667267958188575935810179040280576" ]	a(n) = a(n-1)a(n-2)a(n-3)*a(n-4); for n < 5, a(n) = n.
A000337	[ "0", "1", "5", "17", "49", "129", "321", "769", "1793", "4097", "9217", "20481", "45057", "98305", "212993", "458753", "983041", "2097153", "4456449", "9437185", "19922945", "41943041", "88080385", "184549377", "385875969", "805306369", "1677721601", "3489660929", "7247757313", "15032385537", "31138512897", "64424509441" ]	a(n) = (n-1)*2^n + 1.
A000338	[ "5", "18", "42", "75", "117", "168", "228", "297", "375", "462", "558", "663", "777", "900", "1032", "1173", "1323", "1482", "1650", "1827", "2013", "2208", "2412", "2625", "2847", "3078", "3318", "3567", "3825", "4092", "4368", "4653", "4947", "5250", "5562", "5883", "6213", "6552", "6900", "7257", "7623", "7998", "8382", "8775", "9177", "9588", "10008", "10437", "10875", "11322", "11778" ]	Expansion of x^3(5-2x)*(1-x^3)/(1-x)^4.
A000339	[ "1", "5", "18", "45", "100", "185", "323", "522", "804", "1180", "1687", "2322", "3139", "4146", "5377", "6859", "8645", "10733", "13203", "16058", "19356", "23132", "27460", "32330", "37846", "44031", "50954", "58637", "67203", "76613", "87021", "98443", "110951", "124616", "139526", "155681", "173246", "192243" ]	Number of partitions into non-integral powers.
A000340	[ "1", "5", "18", "58", "179", "543", "1636", "4916", "14757", "44281", "132854", "398574", "1195735", "3587219", "10761672", "32285032", "96855113", "290565357", "871696090", "2615088290", "7845264891", "23535794695", "70607384108", "211822152348", "635466457069" ]	a(0)=1, a(n) = 3*a(n-1) + n + 1.
A000341	[ "1", "2", "3", "6", "26", "96", "210", "1106", "3759", "12577", "74072", "423884", "2333828", "16736611", "99838851", "630091746", "4525325020", "38848875650", "342245714017", "3335164762941", "31315463942337", "241353231085002", "2350106537365732", "17903852593938447", "158065352670318614", "1815064841856534244", "20577063085601738871", "276081763499377227299" ]	Number of ways to pair up {1..2n} so sum of each pair is prime.
A000342	[ "0", "0", "0", "0", "0", "1", "5", "19", "61", "180", "498", "1323", "3405", "8557", "21103", "51248", "122898", "291579", "685562", "1599209", "3705122", "8532309", "19543867", "44552066", "101124867", "228640542", "515125815", "1156829459", "2590247002", "5784031485", "12883390590", "28629914457" ]	Number of n-node rooted trees of height 5.
A000343	[ "1", "5", "20", "70", "230", "721", "2200", "6575", "19385", "56575", "163952", "472645", "1357550", "3888820", "11119325", "31753269", "90603650", "258401245", "736796675", "2100818555", "5990757124", "17087376630", "48753542665", "139155765455", "397356692275", "1135163887190", "3244482184720", "9277856948255" ]	5th power of rooted tree enumerator; number of linear forests of 5 rooted trees.
A000344	[ "1", "5", "20", "75", "275", "1001", "3640", "13260", "48450", "177650", "653752", "2414425", "8947575", "33266625", "124062000", "463991880", "1739969550", "6541168950", "24647883000", "93078189750", "352207870014", "1335293573130", "5071418015120", "19293438101000", "73514652074500", "280531912316292" ]	a(n) = 5*binomial(2n, n-2)/(n+3).
A000345	[ "1", "5", "22", "71", "186", "427", "888", "1704", "3053", "5203", "8476", "13318", "20265", "29946", "43254", "61171", "84832", "115713", "155382", "205779", "269065", "347906", "445001", "563685", "707637", "881042", "1088339", "1335019", "1626233", "1968701", "2369320", "2835467", "3375820", "3999234", "4715586", "5535965", "6472005", "7536195", "8742102", "10105163", "11640190", "13365254", "15298155", "17458190", "19866739", "22546131", "25519743", "28813410", "32453730", "36469433", "40890672", "45749944", "51081147", "56919908", "63304577", "70275008", "77873381", "86145156", "95134772", "104893757", "115473250", "126926418", "139311512", "152687434", "167115830", "182663928", "199398527", "217392226", "236717247", "257454630", "279683011", "303488723", "328959602", "356186407" ]	Number of partitions into non-integral powers.
A000346	[ "1", "5", "22", "93", "386", "1586", "6476", "26333", "106762", "431910", "1744436", "7036530", "28354132", "114159428", "459312152", "1846943453", "7423131482", "29822170718", "119766321572", "480832549478", "1929894318332", "7744043540348", "31067656725032", "124613686513778", "499744650202436" ]	a(n) = 2^(2n+1) - binomial(2n+1, n+1).
A000347	[ "1", "5", "24", "84", "251", "653", "1543", "3341", "6763", "12879", "23446", "40883", "68757", "111976", "177358", "273926", "413784", "612430", "889959", "1271709", "1789841", "2483779", "3402623", "4605954", "6166614", "8171174", "10724604", "13950011", "17994136", "23029141", "29255902", "36908235", "46257694", "57616522", "71344257", "87853381", "107612397" ]	Number of partitions into non-integral powers.
A000348	[ "1", "1", "2", "4", "12", "9", "72", "160", "428", "2434", "3011", "10337", "126962", "264182", "783550", "5004266", "34340141", "176302123", "1188146567", "4457147441", "7845512385", "132253267889", "1004345333251", "3865703506342", "40719018858150", "213982561376958", "1266218151414286", "10976172953868304", "59767467676582641", "512279001476451101", "6189067229056357433" ]	Number of ways to pair up {1^2, 2^2, ..., (2n)^2 } so sum of each pair is prime.
A000349	[ "0", "0", "0", "1", "5", "24", "128", "835", "6423", "56410", "554306", "6016077", "71426225", "920484892", "12793635300", "190730117959", "3035659077083", "51371100102990", "920989078354838", "17437084517068465", "347647092476801301", "7280060180210901232", "159755491837445900120", "3665942433747225901707" ]	One-half the number of permutations of length n with exactly 2 rising or falling successions.
A000350	[ "0", "1", "5", "25", "29", "41", "49", "61", "65", "85", "89", "101", "125", "145", "149", "245", "265", "365", "385", "485", "505", "601", "605", "625", "649", "701", "725", "745", "749", "845", "865", "965", "985", "1105", "1205", "1249", "1345", "1445", "1585", "1685", "1825", "1925", "2065", "2165", "2305", "2405", "2501", "2545", "2645", "2785", "2885" ]	Numbers m such that Fibonacci(m) ends with m.
A000351	[ "1", "5", "25", "125", "625", "3125", "15625", "78125", "390625", "1953125", "9765625", "48828125", "244140625", "1220703125", "6103515625", "30517578125", "152587890625", "762939453125", "3814697265625", "19073486328125", "95367431640625", "476837158203125", "2384185791015625", "11920928955078125" ]	Powers of 5: a(n) = 5^n.
A000352	[ "5", "29", "118", "418", "1383", "4407", "13736", "42236", "128761", "390385", "1179354", "3554454", "10696139", "32153963", "96592972", "290041072", "870647517", "2612991141", "7841070590", "23527406090", "70590606895", "211788597919", "635399348208", "1906265153508", "5718929678273", "17157057470297" ]	One half of the number of permutations of [n] such that the differences have three runs with the same signs.
A000353	[ "7", "23", "47", "59", "167", "179", "263", "383", "503", "863", "887", "983", "1019", "1367", "1487", "1619", "1823", "2063", "2099", "2207", "2447", "2459", "2579", "2819", "2903", "3023", "3167", "3623", "3779", "3863", "4007", "4127", "4139", "4259", "4703", "5087", "5099", "5807", "5927", "5939", "6047", "6659", "6779", "6899", "6983" ]	Primes == 7, 19, 23 (mod 40) such that (p-1)/2 is also prime.
A000354	[ "1", "1", "5", "29", "233", "2329", "27949", "391285", "6260561", "112690097", "2253801941", "49583642701", "1190007424825", "30940193045449", "866325405272573", "25989762158177189", "831672389061670049", "28276861228096781665", "1017967004211484139941", "38682746160036397317757" ]	Expansion of e.g.f. exp(-x)/(1-2*x).
A000355	[ "3", "11", "23", "29", "83", "89", "131", "191", "251", "431", "443", "491", "509", "683", "743", "809", "911", "1031", "1049", "1103", "1223", "1229", "1289", "1409", "1451", "1511", "1583", "1811", "1889", "1931", "2003", "2063", "2069", "2129", "2351", "2543", "2549", "2903", "2963", "2969", "3023", "3329", "3389", "3449", "3491", "3623", "3803" ]	Primes = 3, 9, 11 (mod 20) such that 2p+1 is also prime.
A000356	[ "1", "5", "35", "294", "2772", "28314", "306735", "3476330", "40831076", "493684828", "6114096716", "77266057400", "993420738000", "12964140630900", "171393565105575", "2291968851019650", "30961684478686500", "422056646314726500" ]	Number of rooted cubic maps with 2n nodes and a distinguished Hamiltonian cycle: (2n)!(2n+1)! / (n!^2*(n+1)!(n+2)!).
A000357	[ "1", "1", "5", "35", "315", "3455", "44590", "660665", "11035095", "204904830", "4183174520", "93055783320", "2238954627848", "57903797748386", "1601122732128779", "47120734323344439", "1470076408565099152", "48449426629560437576", "1681560512531504058350", "61293054886119796799892" ]	Number of 5-level labeled rooted trees with n leaves.
A000358	[ "1", "2", "2", "3", "3", "5", "5", "8", "10", "15", "19", "31", "41", "64", "94", "143", "211", "329", "493", "766", "1170", "1811", "2787", "4341", "6713", "10462", "16274", "25415", "39651", "62075", "97109", "152288", "238838", "375167", "589527", "927555", "1459961", "2300348", "3626242", "5721045", "9030451", "14264309", "22542397", "35646312", "56393862", "89264835", "141358275" ]	Number of binary necklaces of length n with no subsequence 00, excluding the necklace "0".
A000359	[ "1", "5", "40", "440", "6170", "105315", "2120610", "49242470", "1296133195", "38152216495", "1242274374380", "44345089721923", "1722416374173854", "72330102999829054", "3265871028909088036", "157797437377747327987", "8124524883679977475839", "444098724261935142753430" ]	Coefficients of iterated exponentials.
A000360	[ "1", "0", "1", "1", "1", "1", "2", "0", "2", "2", "2", "1", "3", "1", "2", "1", "2", "2", "4", "1", "4", "3", "3", "1", "4", "2", "4", "2", "3", "2", "3", "0", "3", "3", "4", "2", "6", "3", "5", "2", "5", "4", "7", "2", "6", "4", "4", "1", "5", "3", "6", "3", "6", "4", "6", "1", "5", "4", "5", "2", "5", "2", "3", "1", "3", "3", "6", "2", "7", "5", "6", "2", "8", "5", "9", "4", "8", "5", "7", "1", "7", "6", "9", "4", "11", "6", "9", "3", "8", "6", "10", "3", "8", "5", "5", "1", "6", "4", "8", "4", "9", "6", "9", "2" ]	Distribution of nonempty triangles inside a fractal rep-4-tile.
A000361	[ "1", "0", "2", "1", "1", "2", "5", "0", "10", "6", "3", "2", "19", "2", "10", "1", "5", "10", "89", "1", "170", "28", "7", "2", "71", "12", "170", "5", "25", "10", "21", "0", "42", "26", "51", "10", "1251", "38", "682", "6", "301", "170", "5833", "3", "2730", "120", "15", "2", "271", "56" ]	From a fractal set of positive Lebesgue measure, a self-replicating tiling with holes, the 4-reptile following the 2-reptile of Paul Levy.
A000362	[ "5", "57", "352", "1280", "3522", "7970", "15872", "29184", "49410", "79042", "122400", "180224", "257314", "362340", "492032", "655360", "867588", "1117314", "1420320", "1803264", "2237380", "2745154", "3380736", "4080640", "4881250", "5874150", "6928416", "8126464", "9600870", "11133604" ]	Generalized class numbers c_(n,2).
A000363	[ "5", "61", "479", "3111", "18270", "101166", "540242", "2819266", "14494859", "73802835", "373398489", "1881341265", "9453340172", "47417364268", "237571096820", "1189405165908", "5951965440609", "29775517732665", "148927275340835", "744793282001995" ]	Number of permutations of [n] with exactly 2 increasing runs of length at least 2.
A000364	[ "1", "1", "5", "61", "1385", "50521", "2702765", "199360981", "19391512145", "2404879675441", "370371188237525", "69348874393137901", "15514534163557086905", "4087072509293123892361", "1252259641403629865468285", "441543893249023104553682821", "177519391579539289436664789665" ]	Euler (or secant or "Zig") numbers: e.g.f. (even powers only) sec(x) = 1/cos(x).
A000365	[ "5", "93", "1030", "8885", "65954", "442610", "2762412", "16322085", "92400330", "505403910", "2687477780", "13957496098", "71053094420", "355548314180", "1752827693528", "8529176056965", "41026491589722", "195327793313790", "921451498774660", "4311086414580022", "20019238138410940" ]	Number of genus 0 rooted planar maps with 4 faces and n vertices.
A000366	[ "1", "1", "2", "7", "38", "295", "3098", "42271", "726734", "15366679", "391888514", "11860602415", "420258768950", "17233254330343", "809698074358250", "43212125903877439", "2599512037272630686", "175079893678534943287", "13122303354155987156306" ]	Genocchi numbers of second kind (A005439) divided by 2^(n-1).
A000367	[ "1", "1", "-1", "1", "-1", "5", "-691", "7", "-3617", "43867", "-174611", "854513", "-236364091", "8553103", "-23749461029", "8615841276005", "-7709321041217", "2577687858367", "-26315271553053477373", "2929993913841559", "-261082718496449122051" ]	Numerators of Bernoulli numbers B_2n.
A000368	[ "1", "1", "4", "9", "28", "71", "202", "542", "1507", "4114", "11381", "31349", "86845", "240567", "668553", "1860361", "5188767", "14495502", "40572216", "113743293", "319405695", "898288484", "2530058013", "7135848125", "20152898513", "56986883801" ]	Number of connected graphs with one cycle of length 4.
A000369	[ "1", "3", "1", "21", "9", "1", "231", "111", "18", "1", "3465", "1785", "345", "30", "1", "65835", "35595", "7650", "825", "45", "1", "1514205", "848925", "196245", "24150", "1680", "63", "1", "40883535", "23586255", "5755050", "775845", "62790", "3066", "84", "1", "1267389585", "748471185", "190482705", "27478710" ]	Triangle of numbers related to triangle A049213; generalization of Stirling numbers of second kind A008277, Bessel triangle A001497.
A000370	[ "1", "2", "4", "14", "222", "616126", "200253952527184", "263735716028826576482466871188128", "5609038300883759793482640992086670939164957990135057216103303119630336" ]	Number of NPN-equivalence classes of Boolean functions of n or fewer variables.
A000371	[ "2", "2", "10", "218", "64594", "4294642034", "18446744047940725978", "340282366920938463334247399005993378250", "115792089237316195423570985008687907850547725730273056332267095982282337798562" ]	a(n) = Sum_{k=0..n} (-1)^(n-k)binomial(n,k)2^(2^k).
A000372	[ "2", "3", "6", "20", "168", "7581", "7828354", "2414682040998", "56130437228687557907788" ]	Dedekind numbers or Dedekind's problem: number of monotone Boolean functions of n variables, number of antichains of subsets of an n-set, number of elements in a free distributive lattice on n generators, number of Sperner families.
A000373	[ "0", "0", "1", "8", "44", "214", "1000", "4592", "20888", "94846", "434973", "2042836", "9979086", "51460622", "283839957", "1688139424", "10859199656", "75338888918", "560740210491", "4445766353604", "37329808482989", "330143634313064", "3064464030121369" ]	Conjectured dimension of a module associated with the free commutative Moufang loop with n generators.
A000374	[ "1", "1", "2", "1", "2", "2", "3", "1", "3", "2", "2", "2", "2", "3", "5", "1", "3", "3", "2", "2", "6", "2", "3", "2", "3", "2", "4", "3", "2", "5", "7", "1", "5", "3", "6", "3", "2", "2", "5", "2", "3", "6", "4", "2", "8", "3", "3", "2", "5", "3", "8", "2", "2", "4", "5", "3", "5", "2", "2", "5", "2", "7", "13", "1", "7", "5", "2", "3", "6", "6", "3", "3", "9", "2", "8", "2", "6", "5", "3", "2", "5", "3", "2", "6", "12", "4", "5", "2", "9", "8", "10", "3", "14", "3", "5", "2", "3", "5", "8", "3" ]	Number of cycles (mod n) under doubling map.
A000375	[ "0", "1", "2", "4", "7", "10", "16", "22", "30", "38", "51", "65", "80", "101", "113", "139", "159", "191", "221" ]	Topswops (1): start by shuffling n cards labeled 1..n. If top card is m, reverse order of top m cards, then repeat. a(n) is the maximal number of steps before top card is 1.
A000376	[ "0", "1", "2", "4", "7", "10", "16", "22", "30", "38", "51", "63", "80", "101", "112", "130", "159", "191", "207", "231" ]	Topswops (2): start by shuffling n cards labeled 1..n. If top card is m, reverse order of top m cards. Repeat until 1 gets to top, then stop. Suppose the whole deck is now sorted (if not, discard this case). a(n) is the maximal number of steps before 1 got to the top.
A000377	[ "1", "1", "1", "1", "1", "2", "1", "2", "1", "1", "2", "2", "1", "0", "2", "2", "1", "0", "1", "0", "2", "2", "2", "0", "1", "3", "0", "1", "2", "2", "2", "2", "1", "2", "0", "4", "1", "0", "0", "0", "2", "0", "2", "0", "2", "2", "0", "0", "1", "3", "3", "0", "0", "2", "1", "4", "2", "0", "2", "2", "2", "0", "2", "2", "1", "0", "2", "0", "0", "0", "4", "0", "1", "2", "0", "3", "0", "4", "0", "2", "2", "1", "0", "2", "2", "0", "0", "2", "2", "0", "2", "0", "0", "2", "0", "0", "1", "2", "3", "2", "3", "2" ]	Expansion of f(-q^3) * f(-q^8) * chi(-q^12) / chi(-q) in powers of q where chi(), f() are Ramanujan theta functions.
A000378	[ "0", "1", "2", "3", "4", "5", "6", "8", "9", "10", "11", "12", "13", "14", "16", "17", "18", "19", "20", "21", "22", "24", "25", "26", "27", "29", "30", "32", "33", "34", "35", "36", "37", "38", "40", "41", "42", "43", "44", "45", "46", "48", "49", "50", "51", "52", "53", "54", "56", "57", "58", "59", "61", "62", "64", "65", "66", "67", "68", "69", "70", "72", "73", "74", "75", "76", "77", "78", "80", "81", "82", "83" ]	Sums of three squares: numbers of the form x^2 + y^2 + z^2.
A000379	[ "1", "6", "8", "10", "12", "14", "15", "18", "20", "21", "22", "26", "27", "28", "32", "33", "34", "35", "36", "38", "39", "44", "45", "46", "48", "50", "51", "52", "55", "57", "58", "62", "63", "64", "65", "68", "69", "74", "75", "76", "77", "80", "82", "85", "86", "87", "91", "92", "93", "94", "95", "98", "99", "100", "106", "111", "112", "115", "116", "117", "118", "119", "120", "122", "123", "124", "125", "129" ]	Numbers n where total number of 1-bits in the exponents of their prime factorization is even; a 2-way classification of integers: complement of A000028.
A000380	[ "6", "8", "40", "176", "1421", "10352", "93114", "912920", "9929997", "117970704", "1521176826", "21150414880", "315400444070", "5020920314016", "84979755347122", "1523710321272384", "28851091193764023", "575253584489378040", "12047084261153160394", "264377395040950523112", "6066972656940255290199" ]	Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-3 places.
A000381	[ "2", "3", "4", "6", "9", "14", "22", "35", "56", "90", "145", "234", "378", "611", "988", "1598", "2585", "4182", "6766" ]	Essentially the same as A001611.
A000382	[ "6", "11", "20", "36", "65", "119", "218", "400", "735", "1351", "2484", "4568", "8401", "15451", "28418", "52268", "96135", "176819", "325220", "598172", "1100209", "2023599", "3721978", "6845784", "12591359", "23159119", "42596260", "78346736", "144102113", "265045107", "487493954" ]	Restricted permutations.
A000383	[ "1", "1", "1", "1", "1", "1", "6", "11", "21", "41", "81", "161", "321", "636", "1261", "2501", "4961", "9841", "19521", "38721", "76806", "152351", "302201", "599441", "1189041", "2358561", "4678401", "9279996", "18407641", "36513081", "72426721", "143664401", "284970241", "565262081", "1121244166", "2224080691", "4411648301" ]	Hexanacci numbers with a(0) = ... = a(5) = 1.
A000384	[ "0", "1", "6", "15", "28", "45", "66", "91", "120", "153", "190", "231", "276", "325", "378", "435", "496", "561", "630", "703", "780", "861", "946", "1035", "1128", "1225", "1326", "1431", "1540", "1653", "1770", "1891", "2016", "2145", "2278", "2415", "2556", "2701", "2850", "3003", "3160", "3321", "3486", "3655", "3828", "4005", "4186", "4371", "4560" ]	Hexagonal numbers: a(n) = n(2n-1).
A000385	[ "1", "6", "17", "38", "70", "116", "185", "258", "384", "490", "686", "826", "1124", "1292", "1705", "1896", "2491", "2670", "3416", "3680", "4602", "4796", "6110", "6178", "7700", "7980", "9684", "9730", "12156", "11920", "14601", "14752", "17514", "17224", "21395", "20406", "24590", "24556", "28920", "27860", "34112", "32186", "38674", "37994", "43980", "42136", "51646", "47772", "56749", "55500", "64316", "60606", "73420", "67956", "80500", "77760", "88860", "83810", "102284", "92690", "108752", "105236", "120777", "112672", "135120", "123046", "145194", "138656", "157512", "146580", "177515", "159396", "185744", "179122" ]	Convolution of A000203 with itself.
A000386	[ "0", "0", "0", "1", "6", "20", "134", "915", "7324", "65784", "657180", "7223637", "86637650", "1125842556", "15757002706", "236298742375", "3780061394232", "64251145312880", "1156374220457784", "21968796934412649", "439337048505773790", "9225384943965382564", "202945418255342821470" ]	Coefficients of ménage hit polynomials.
A000387	[ "0", "0", "1", "0", "6", "20", "135", "924", "7420", "66744", "667485", "7342280", "88107426", "1145396460", "16035550531", "240533257860", "3848532125880", "65425046139824", "1177650830516985", "22375365779822544", "447507315596451070", "9397653627525472260", "206748379805560389951" ]	Rencontres numbers: number of permutations of [n] with exactly two fixed points.
A000388	[ "6", "20", "180", "1106", "9292", "82980", "831545", "9139482", "109595496", "1423490744", "19911182207", "298408841160", "4770598226296", "81037124739588", "1457607971046492", "27675791180024802", "553166885187641670", "11609691036091870428", "255273744004170486155", "5868308906885934514178" ]	Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-2 places.
A000389	[ "0", "0", "0", "0", "0", "1", "6", "21", "56", "126", "252", "462", "792", "1287", "2002", "3003", "4368", "6188", "8568", "11628", "15504", "20349", "26334", "33649", "42504", "53130", "65780", "80730", "98280", "118755", "142506", "169911", "201376", "237336", "278256", "324632", "376992", "435897", "501942", "575757", "658008", "749398" ]	Binomial coefficients C(n,5).
A000390	[ "1", "6", "21", "71", "216", "657", "1907", "5507", "15522", "43352", "119140", "323946", "869476", "2308071", "6056581", "15724170", "40393693", "102736274", "258790004", "645968054", "1598460229", "3923114261", "9554122089", "23098084695", "55458417125", "132293945737", "313657570114" ]	Number of 5-dimensional partitions of n.
A000391	[ "1", "6", "21", "71", "216", "672", "1982", "5817", "16582", "46633", "128704", "350665", "941715", "2499640", "6557378", "17024095", "43756166", "111433472", "281303882", "704320180", "1749727370", "4314842893", "10565857064", "25700414815", "62115621317", "149214574760", "356354881511", "846292135184" ]	Euler transform of A000332.
A000392	[ "0", "0", "0", "1", "6", "25", "90", "301", "966", "3025", "9330", "28501", "86526", "261625", "788970", "2375101", "7141686", "21457825", "64439010", "193448101", "580606446", "1742343625", "5228079450", "15686335501", "47063200806", "141197991025", "423610750290", "1270865805301" ]	Stirling numbers of second kind S(n,3).
A000393	[ "0", "0", "0", "0", "0", "0", "1", "6", "26", "94", "308", "941", "2744", "7722", "21166", "56809", "149971", "390517", "1005491", "2564164", "6485901", "16289602", "40659669", "100934017", "249343899", "613286048", "1502515487", "3667953650", "8925161513", "21652815724", "52387028291" ]	Number of n-node rooted trees of height 6.
A000394	[ "0", "1", "2", "4", "5", "7", "8", "9", "10", "11", "12", "13", "15", "16", "17", "18", "20", "23", "24", "25", "26", "27", "28", "29", "30", "32", "33", "34", "36", "37", "38", "39", "40", "41", "43", "44", "45", "46", "47", "48", "49", "50", "52", "53", "54", "56", "57", "58", "59", "60", "61", "62", "63", "64", "65", "67", "68", "69", "71", "72" ]	Numbers of form x^2 + y^2 + 7z^2.
A000395	[ "1", "6", "27", "104", "369", "1236", "3989", "12522", "38535", "116808", "350064", "1039896", "3068145", "9004182", "26314773", "76652582", "222705603", "645731148", "1869303857", "5404655358", "15611296146", "45060069406", "129989169909", "374843799786", "1080624405287" ]	6th power of rooted tree enumerator; number of linear forests of 6 rooted trees.
A000396	[ "6", "28", "496", "8128", "33550336", "8589869056", "137438691328", "2305843008139952128", "2658455991569831744654692615953842176", "191561942608236107294793378084303638130997321548169216" ]	Perfect numbers k: k is equal to the sum of the proper divisors of k.
A000397	[ "6", "32", "109", "288", "654", "1337", "2506", "4414", "7379", "11822", "18273", "27356", "39938", "56974", "79607", "109267", "147523", "196295", "257715", "334407", "429086", "545034", "685917", "855886", "1059360", "1301776", "1588321", "1925620", "2320544", "2780468", "3314007", "3930001", "4638319", "5449943", "6376505", "7430471", "8625369", "9976540", "11498855", "13210238", "15128487", "17272896", "19664754", "22326319", "25280987", "28554486", "32173404", "36166409", "40563607", "45397395", "50701682", "56512012", "62866699", "69805531", "77370606", "85607286", "94560129", "104280410", "114819255", "126229853", "138570284", "151899428", "166278945", "181775849", "198456941", "216394746", "235661505", "256338017", "278503009", "302242623", "327644632", "354799834", "383805368", "414759214", "447764499", "482931051" ]	Number of partitions into non-integral powers.
A000398	[ "0", "1", "2", "3", "4", "5", "6", "8", "9", "11", "12", "13", "14", "15", "16", "17", "18", "19", "20", "22", "23", "24", "25", "26", "27", "28", "29", "30", "31", "32", "33", "34", "36", "37", "38", "39", "40", "41", "43", "44", "45", "46", "47", "48", "49", "50", "51", "52", "53", "54", "55", "56", "57", "58", "59", "60", "61", "62", "63", "64" ]	Numbers of form x^2 + 2y^2 + 2yz + 4z^2.
A000399	[ "1", "6", "35", "225", "1624", "13132", "118124", "1172700", "12753576", "150917976", "1931559552", "26596717056", "392156797824", "6165817614720", "102992244837120", "1821602444624640", "34012249593822720", "668609730341153280", "13803759753640704000" ]	Unsigned Stirling numbers of first kind s(n,3).
A000400	[ "1", "6", "36", "216", "1296", "7776", "46656", "279936", "1679616", "10077696", "60466176", "362797056", "2176782336", "13060694016", "78364164096", "470184984576", "2821109907456", "16926659444736", "101559956668416", "609359740010496", "3656158440062976", "21936950640377856", "131621703842267136" ]	Powers of 6: a(n) = 6^n.

A000301

[ "1", "2", "2", "4", "8", "32", "256", "8192", "2097152", "17179869184", "36028797018963968", "618970019642690137449562112", "22300745198530623141535718272648361505980416", "13803492693581127574869511724554050904902217944340773110325048447598592" ]

a(n) = a(n-1)*a(n-2) with a(0) = 1, a(1) = 2; also a(n) = 2^Fibonacci(n).

A000302

[ "1", "4", "16", "64", "256", "1024", "4096", "16384", "65536", "262144", "1048576", "4194304", "16777216", "67108864", "268435456", "1073741824", "4294967296", "17179869184", "68719476736", "274877906944", "1099511627776", "4398046511104", "17592186044416", "70368744177664", "281474976710656" ]

Powers of 4: a(n) = 4^n.

A000303

[ "0", "1", "4", "16", "69", "348", "2016", "13357", "99376", "822040", "7477161", "74207208", "797771520", "9236662345", "114579019468", "1516103040832", "21314681315997", "317288088082404", "4985505271920096", "82459612672301845", "1432064398910663704", "26054771465540507272" ]

Number of permutations of [n] in which the longest increasing run has length 2.

A000304

[ "2", "3", "6", "18", "108", "1944", "209952", "408146688", "85691213438976", "34974584955819144511488", "2997014624388697307377363936018956288", "104819342594514896999066634490728502944926883876041385836544" ]

a(n) = a(n-1)*a(n-2).

A000305

[ "1", "4", "18", "89", "466", "2537", "14209", "81316", "473338", "2793454", "16674417", "100487896", "610549829", "3735850007", "23000055178", "142370597601", "885521350882", "5531501612071", "34686798239678", "218273864005214", "1377897874711437" ]

Number of certain rooted planar maps.

A000306

[ "1", "4", "19", "66", "219", "645", "1813", "4802", "12265", "30198", "72396", "169231", "387707", "871989", "1930868", "4215615", "9091410", "19389327", "40944999", "85691893", "177898521" ]

Number of trees of diameter 8.

A000307

[ "1", "1", "4", "22", "154", "1304", "12915", "146115", "1855570", "26097835", "402215465", "6734414075", "121629173423", "2355470737637", "48664218965021", "1067895971109199", "24795678053493443", "607144847919796830", "15630954703539323090", "421990078975569031642", "11918095123121138408128" ]

Number of 4-level labeled rooted trees with n leaves.

A000308

[ "1", "2", "3", "6", "36", "648", "139968", "3265173504", "296148833645101056", "135345882205792807436868315512832", "130876399105969522361889021452224949874232743897657526714368" ]

a(n) = a(n-1)*a(n-2)*a(n-3) with a(1)=1, a(2)=2 and a(3)=3.

A000309

[ "1", "1", "4", "24", "176", "1456", "13056", "124032", "1230592", "12629760", "133186560", "1436098560", "15774990336", "176028860416", "1990947110912", "22783499599872", "263411369705472", "3073132646563840", "36143187370967040", "428157758086840320", "5105072641718353920", "61228492804372561920" ]

Number of rooted planar bridgeless cubic maps with 2n nodes.

A000310

[ "1", "4", "26", "234", "2696", "37919", "630521", "12111114", "264051201", "6445170229", "174183891471", "5164718385337", "166737090160871", "5822980248613990", "218756388226681557", "8797723991458469015", "377159237609540937788", "17170729962232112834302", "827382365085791968518198", "42070004707327023844695198" ]

Coefficients of iterated exponentials.

A000311

[ "0", "1", "1", "4", "26", "236", "2752", "39208", "660032", "12818912", "282137824", "6939897856", "188666182784", "5617349020544", "181790703209728", "6353726042486272", "238513970965257728", "9571020586419012608", "408837905660444010496", "18522305410364986906624" ]

Schroeder's fourth problem; also series-reduced rooted trees with n labeled leaves; also number of total partitions of n.

A000312

[ "1", "1", "4", "27", "256", "3125", "46656", "823543", "16777216", "387420489", "10000000000", "285311670611", "8916100448256", "302875106592253", "11112006825558016", "437893890380859375", "18446744073709551616", "827240261886336764177", "39346408075296537575424", "1978419655660313589123979" ]

a(n) = n^n; number of labeled mappings from n points to themselves (endofunctions).

A000313

[ "0", "0", "0", "1", "4", "30", "220", "1855", "17304", "177996", "2002440", "24474285", "323060540", "4581585866", "69487385604", "1122488536715", "19242660629360", "348933579412440", "6673354706262864", "134252194678935321", "2834212998777523380", "62651024183503148470", "1447238658638922729580" ]

Number of permutations of length n with 3 consecutive ascending pairs.

A000314

[ "1", "1", "1", "4", "31", "362", "5676", "111982", "2666392", "74433564", "2384579440", "86248530296", "3476794472064", "154579941792256", "7514932528712896", "396595845237540600", "22581060079942183936", "1379771773100463174608", "90059660791562688208128", "6253914166368448348512064" ]

Number of mixed Husimi trees with n nodes; or labeled polygonal cacti with bridges.

A000315

[ "1", "1", "1", "4", "56", "9408", "16942080", "535281401856", "377597570964258816", "7580721483160132811489280", "5363937773277371298119673540771840" ]

Number of reduced Latin squares of order n; also number of labeled loops (quasigroups with an identity element) with a fixed identity element.

A000316

[ "1", "0", "4", "80", "4752", "440192", "59245120", "10930514688", "2649865335040", "817154768973824", "312426715251262464", "145060238642780180480", "80403174342119992692736", "52443098500204184915312640", "39764049487996490505336537088" ]

Two decks each have n kinds of cards, 2 of each kind. The first deck is laid out in order. The second deck is shuffled and laid out next to the first. A match occurs if a card from the second deck is next to a card of the same kind from the first deck. a(n) is the number of ways of achieving no matches.

A000317

[ "1", "2", "3", "7", "37", "1159", "1301767", "1693089917617", "2866551265129451657751739", "8217116155610406522540626640615749228405055996847" ]

a(n+1) = a(n)^2 - a(n) a(n-1) + a(n-1)^2.

A000318

[ "4", "128", "16384", "4456448", "2080374784", "1483911200768", "1501108249821184", "2044143848640217088", "3605459138582973251584", "7995891855149741436305408", "21776918737280678860353961984", "71454103701490016776039304265728", "278008871543597996197497752082448384" ]

Generalized tangent numbers d(4,n).

A000319

[ "1", "1", "74", "-1", "-2", "-3", "0", "1", "30", "-2", "-2", "29", "1", "4", "-6", "0", "1", "2", "-1", "-1", "-1", "-1", "-2", "-9", "0", "0", "1", "2", "-2", "-35", "-1", "-1", "-1", "-1", "-1", "-1", "-1", "-2", "-3", "0", "0", "1", "5", "-2", "-2", "3", "1", "1", "-4", "-1", "-1", "-1", "-1", "-1", "-1", "-1", "-1", "-2", "-3", "1", "2", "-1", "-1", "-1", "-1", "-1", "-1", "-1", "-1", "-1", "-1", "-1", "-1", "-1", "-1", "-1", "-1", "-1", "-1", "-2", "-3", "0", "1", "2", "-1", "-2", "-21", "-7", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0" ]

a(n) = floor(b(n)), where b(n) = tan(b(n-1)), b(0)=1.

A000320

[ "4", "272", "55744", "23750912", "17328937984", "19313964388352", "30527905292468224", "64955605537174126592", "179013508069217017790464", "620314831396713435870789632", "2639743384489464189324523208704", "13533573366345611477262311433961472", "82274260343572247169162187576069586944" ]

Generalized tangent numbers d(5,n).

A000321

[ "1", "-1", "-1", "5", "1", "-41", "31", "461", "-895", "-6481", "22591", "107029", "-604031", "-1964665", "17669471", "37341149", "-567425279", "-627491489", "19919950975", "2669742629", "-759627879679", "652838174519", "31251532771999", "-59976412450835", "-1377594095061119", "4256461892701199", "64623242860354751" ]

H_n(-1/2), where H_n(x) is Hermite polynomial of degree n.

A000322

[ "1", "1", "1", "1", "1", "5", "9", "17", "33", "65", "129", "253", "497", "977", "1921", "3777", "7425", "14597", "28697", "56417", "110913", "218049", "428673", "842749", "1656801", "3257185", "6403457", "12588865", "24749057", "48655365", "95653929", "188050673", "369697889", "726806913", "1428864769" ]

Pentanacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) + a(n-5) with a(0) = a(1) = a(2) = a(3) = a(4) = 1.

A000323

[ "5", "9", "21", "37", "69", "69", "89", "137", "177", "421", "481", "657", "749", "885", "1085", "1305", "1353", "1489", "1861", "2617", "2693", "3125", "5249", "5761", "7129", "8109", "9465", "9465", "10717", "12401", "12401", "16237", "16237", "24833", "30725", "35237", "46701", "47441", "47441", "61493", "67797", "67805", "67805" ]

Let A(n) = #{(i,j): i^2 + j^2 <= n}, V(n) = Pi*n, P(n) = A(n) - V(n); A000099 gives values of n where |P(n)| sets a new record; sequence gives A(A000099(n)).

A000324

[ "1", "5", "9", "49", "2209", "4870849", "23725150497409", "562882766124611619513723649", "316837008400094222150776738483768236006420971486980609" ]

A nonlinear recurrence: a(0) = 1, a(1) = 5, a(n) = a(n-1)^2 - 4*a(n-1) + 4 for n>1.

A000325

[ "1", "1", "2", "5", "12", "27", "58", "121", "248", "503", "1014", "2037", "4084", "8179", "16370", "32753", "65520", "131055", "262126", "524269", "1048556", "2097131", "4194282", "8388585", "16777192", "33554407", "67108838", "134217701", "268435428", "536870883", "1073741794", "2147483617" ]

a(n) = 2^n - n.

A000326

[ "0", "1", "5", "12", "22", "35", "51", "70", "92", "117", "145", "176", "210", "247", "287", "330", "376", "425", "477", "532", "590", "651", "715", "782", "852", "925", "1001", "1080", "1162", "1247", "1335", "1426", "1520", "1617", "1717", "1820", "1926", "2035", "2147", "2262", "2380", "2501", "2625", "2752", "2882", "3015", "3151" ]

Pentagonal numbers: a(n) = n*(3*n-1)/2.

A000327

[ "1", "5", "12", "23", "39", "62", "91", "127", "171", "228", "294", "370", "461", "561", "677", "811", "955", "1121", "1303", "1499", "1719", "1960", "2218", "2499", "2806", "3131", "3485", "3868", "4274", "4706", "5166", "5658", "6175", "6725", "7309", "7923", "8572", "9256", "9972", "10728", "11521", "12349", "13218", "14126", "15072" ]

Number of partitions into non-integral powers.

A000328

[ "1", "5", "13", "29", "49", "81", "113", "149", "197", "253", "317", "377", "441", "529", "613", "709", "797", "901", "1009", "1129", "1257", "1373", "1517", "1653", "1793", "1961", "2121", "2289", "2453", "2629", "2821", "3001", "3209", "3409", "3625", "3853", "4053", "4293", "4513", "4777", "5025", "5261", "5525", "5789", "6077", "6361", "6625" ]

Number of points of norm <= n^2 in square lattice.

A000329

[ "1", "2", "75", "-1", "-1", "-2", "1", "2", "31", "-1", "-2", "29", "1", "5", "-6", "1", "1", "3", "-1", "-1", "-1", "-1", "-1", "-9", "1", "1", "1", "2", "-2", "-35", "0", "0", "-1", "-1", "-1", "-1", "-1", "-1", "-2", "1", "1", "1", "5", "-1", "-2", "4", "1", "2", "-4", "0", "0", "0", "-1", "-1", "-1", "-1", "-1", "-1", "-2", "1", "3", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "-1" ]

Nearest integer to b(n), where b(n) = tan(b(n-1)), b(0) = 1.

A000330

[ "0", "1", "5", "14", "30", "55", "91", "140", "204", "285", "385", "506", "650", "819", "1015", "1240", "1496", "1785", "2109", "2470", "2870", "3311", "3795", "4324", "4900", "5525", "6201", "6930", "7714", "8555", "9455", "10416", "11440", "12529", "13685", "14910", "16206", "17575", "19019", "20540", "22140", "23821", "25585", "27434", "29370" ]

Square pyramidal numbers: a(n) = 0^2 + 1^2 + 2^2 + ... + n^2 = n*(n+1)*(2*n+1)/6.

A000331

[ "5", "14", "1026", "4324", "311387", "6425694", "579783114", "4028104212", "7315072725560", "61358264615344", "9569450876916944", "1632353370882506848", "1365475358484643531856", "15211641461623992544160", "74766806258361827981250240", "936580261005146914634459520", "6083678228249789825160175706880", "1936651082361926268672618636234240", "688115696843061332335070140230720000", "10517068622936239459488783307672335360", "2913914903970372007778735454555848514846720" ]

Related to zeros of Bessel function.

A000332

[ "0", "0", "0", "0", "1", "5", "15", "35", "70", "126", "210", "330", "495", "715", "1001", "1365", "1820", "2380", "3060", "3876", "4845", "5985", "7315", "8855", "10626", "12650", "14950", "17550", "20475", "23751", "27405", "31465", "35960", "40920", "46376", "52360", "58905", "66045", "73815", "82251", "91390", "101270", "111930", "123410" ]

Binomial coefficient binomial(n,4) = n*(n-1)*(n-2)*(n-3)/24.

A000333

[ "1", "5", "15", "40", "98", "237", "534", "1185", "2554", "5391", "11117", "22556", "44858", "88000", "170107", "324547", "611755", "1140382", "2103554", "3842826", "6955918", "12483075", "22220002", "39248230", "68819781", "119839422", "207304370", "356356801", "608901907", "1034452712", "1747764522", "2937370605", "4911675955", "8173032301" ]

Number of partitions into non-integral powers.

A000334

[ "1", "5", "15", "45", "120", "326", "835", "2145", "5345", "13220", "32068", "76965", "181975", "425490", "982615", "2245444", "5077090", "11371250", "25235790", "55536870", "121250185", "262769080", "565502405", "1209096875", "2569270050", "5427963902", "11404408525", "23836421895", "49573316740", "102610460240" ]

Number of 4-dimensional partitions of n.

A000335

[ "1", "5", "15", "45", "120", "331", "855", "2214", "5545", "13741", "33362", "80091", "189339", "442799", "1023192", "2340904", "5302061", "11902618", "26488454", "58479965", "128120214", "278680698", "602009786", "1292027222", "2755684669", "5842618668", "12317175320", "25825429276", "53865355154", "111786084504", "230867856903", "474585792077", "971209629993" ]

Euler transform of A000292.

A000336

[ "1", "2", "3", "4", "24", "576", "165888", "9172942848", "21035720123168587776", "18437563379178327736384102280592359424", "590180110002114158896983994712576414865667267958188575935810179040280576" ]

a(n) = a(n-1)*a(n-2)*a(n-3)*a(n-4); for n < 5, a(n) = n.

A000337

[ "0", "1", "5", "17", "49", "129", "321", "769", "1793", "4097", "9217", "20481", "45057", "98305", "212993", "458753", "983041", "2097153", "4456449", "9437185", "19922945", "41943041", "88080385", "184549377", "385875969", "805306369", "1677721601", "3489660929", "7247757313", "15032385537", "31138512897", "64424509441" ]

a(n) = (n-1)*2^n + 1.

A000338

[ "5", "18", "42", "75", "117", "168", "228", "297", "375", "462", "558", "663", "777", "900", "1032", "1173", "1323", "1482", "1650", "1827", "2013", "2208", "2412", "2625", "2847", "3078", "3318", "3567", "3825", "4092", "4368", "4653", "4947", "5250", "5562", "5883", "6213", "6552", "6900", "7257", "7623", "7998", "8382", "8775", "9177", "9588", "10008", "10437", "10875", "11322", "11778" ]

Expansion of x^3*(5-2*x)*(1-x^3)/(1-x)^4.

A000339

[ "1", "5", "18", "45", "100", "185", "323", "522", "804", "1180", "1687", "2322", "3139", "4146", "5377", "6859", "8645", "10733", "13203", "16058", "19356", "23132", "27460", "32330", "37846", "44031", "50954", "58637", "67203", "76613", "87021", "98443", "110951", "124616", "139526", "155681", "173246", "192243" ]

Number of partitions into non-integral powers.

A000340

[ "1", "5", "18", "58", "179", "543", "1636", "4916", "14757", "44281", "132854", "398574", "1195735", "3587219", "10761672", "32285032", "96855113", "290565357", "871696090", "2615088290", "7845264891", "23535794695", "70607384108", "211822152348", "635466457069" ]

a(0)=1, a(n) = 3*a(n-1) + n + 1.

A000341

[ "1", "2", "3", "6", "26", "96", "210", "1106", "3759", "12577", "74072", "423884", "2333828", "16736611", "99838851", "630091746", "4525325020", "38848875650", "342245714017", "3335164762941", "31315463942337", "241353231085002", "2350106537365732", "17903852593938447", "158065352670318614", "1815064841856534244", "20577063085601738871", "276081763499377227299" ]

Number of ways to pair up {1..2n} so sum of each pair is prime.

A000342

[ "0", "0", "0", "0", "0", "1", "5", "19", "61", "180", "498", "1323", "3405", "8557", "21103", "51248", "122898", "291579", "685562", "1599209", "3705122", "8532309", "19543867", "44552066", "101124867", "228640542", "515125815", "1156829459", "2590247002", "5784031485", "12883390590", "28629914457" ]

Number of n-node rooted trees of height 5.

A000343

[ "1", "5", "20", "70", "230", "721", "2200", "6575", "19385", "56575", "163952", "472645", "1357550", "3888820", "11119325", "31753269", "90603650", "258401245", "736796675", "2100818555", "5990757124", "17087376630", "48753542665", "139155765455", "397356692275", "1135163887190", "3244482184720", "9277856948255" ]

5th power of rooted tree enumerator; number of linear forests of 5 rooted trees.

A000344

[ "1", "5", "20", "75", "275", "1001", "3640", "13260", "48450", "177650", "653752", "2414425", "8947575", "33266625", "124062000", "463991880", "1739969550", "6541168950", "24647883000", "93078189750", "352207870014", "1335293573130", "5071418015120", "19293438101000", "73514652074500", "280531912316292" ]

a(n) = 5*binomial(2n, n-2)/(n+3).

A000345

[ "1", "5", "22", "71", "186", "427", "888", "1704", "3053", "5203", "8476", "13318", "20265", "29946", "43254", "61171", "84832", "115713", "155382", "205779", "269065", "347906", "445001", "563685", "707637", "881042", "1088339", "1335019", "1626233", "1968701", "2369320", "2835467", "3375820", "3999234", "4715586", "5535965", "6472005", "7536195", "8742102", "10105163", "11640190", "13365254", "15298155", "17458190", "19866739", "22546131", "25519743", "28813410", "32453730", "36469433", "40890672", "45749944", "51081147", "56919908", "63304577", "70275008", "77873381", "86145156", "95134772", "104893757", "115473250", "126926418", "139311512", "152687434", "167115830", "182663928", "199398527", "217392226", "236717247", "257454630", "279683011", "303488723", "328959602", "356186407" ]

Number of partitions into non-integral powers.

A000346

[ "1", "5", "22", "93", "386", "1586", "6476", "26333", "106762", "431910", "1744436", "7036530", "28354132", "114159428", "459312152", "1846943453", "7423131482", "29822170718", "119766321572", "480832549478", "1929894318332", "7744043540348", "31067656725032", "124613686513778", "499744650202436" ]

a(n) = 2^(2*n+1) - binomial(2*n+1, n+1).

A000347

[ "1", "5", "24", "84", "251", "653", "1543", "3341", "6763", "12879", "23446", "40883", "68757", "111976", "177358", "273926", "413784", "612430", "889959", "1271709", "1789841", "2483779", "3402623", "4605954", "6166614", "8171174", "10724604", "13950011", "17994136", "23029141", "29255902", "36908235", "46257694", "57616522", "71344257", "87853381", "107612397" ]

Number of partitions into non-integral powers.

A000348

[ "1", "1", "2", "4", "12", "9", "72", "160", "428", "2434", "3011", "10337", "126962", "264182", "783550", "5004266", "34340141", "176302123", "1188146567", "4457147441", "7845512385", "132253267889", "1004345333251", "3865703506342", "40719018858150", "213982561376958", "1266218151414286", "10976172953868304", "59767467676582641", "512279001476451101", "6189067229056357433" ]

Number of ways to pair up {1^2, 2^2, ..., (2n)^2 } so sum of each pair is prime.

A000349

[ "0", "0", "0", "1", "5", "24", "128", "835", "6423", "56410", "554306", "6016077", "71426225", "920484892", "12793635300", "190730117959", "3035659077083", "51371100102990", "920989078354838", "17437084517068465", "347647092476801301", "7280060180210901232", "159755491837445900120", "3665942433747225901707" ]

One-half the number of permutations of length n with exactly 2 rising or falling successions.

A000350

[ "0", "1", "5", "25", "29", "41", "49", "61", "65", "85", "89", "101", "125", "145", "149", "245", "265", "365", "385", "485", "505", "601", "605", "625", "649", "701", "725", "745", "749", "845", "865", "965", "985", "1105", "1205", "1249", "1345", "1445", "1585", "1685", "1825", "1925", "2065", "2165", "2305", "2405", "2501", "2545", "2645", "2785", "2885" ]

Numbers m such that Fibonacci(m) ends with m.

A000351

[ "1", "5", "25", "125", "625", "3125", "15625", "78125", "390625", "1953125", "9765625", "48828125", "244140625", "1220703125", "6103515625", "30517578125", "152587890625", "762939453125", "3814697265625", "19073486328125", "95367431640625", "476837158203125", "2384185791015625", "11920928955078125" ]

Powers of 5: a(n) = 5^n.

A000352

[ "5", "29", "118", "418", "1383", "4407", "13736", "42236", "128761", "390385", "1179354", "3554454", "10696139", "32153963", "96592972", "290041072", "870647517", "2612991141", "7841070590", "23527406090", "70590606895", "211788597919", "635399348208", "1906265153508", "5718929678273", "17157057470297" ]

One half of the number of permutations of [n] such that the differences have three runs with the same signs.

A000353

[ "7", "23", "47", "59", "167", "179", "263", "383", "503", "863", "887", "983", "1019", "1367", "1487", "1619", "1823", "2063", "2099", "2207", "2447", "2459", "2579", "2819", "2903", "3023", "3167", "3623", "3779", "3863", "4007", "4127", "4139", "4259", "4703", "5087", "5099", "5807", "5927", "5939", "6047", "6659", "6779", "6899", "6983" ]

Primes == 7, 19, 23 (mod 40) such that (p-1)/2 is also prime.

A000354

[ "1", "1", "5", "29", "233", "2329", "27949", "391285", "6260561", "112690097", "2253801941", "49583642701", "1190007424825", "30940193045449", "866325405272573", "25989762158177189", "831672389061670049", "28276861228096781665", "1017967004211484139941", "38682746160036397317757" ]

Expansion of e.g.f. exp(-x)/(1-2*x).

A000355

[ "3", "11", "23", "29", "83", "89", "131", "191", "251", "431", "443", "491", "509", "683", "743", "809", "911", "1031", "1049", "1103", "1223", "1229", "1289", "1409", "1451", "1511", "1583", "1811", "1889", "1931", "2003", "2063", "2069", "2129", "2351", "2543", "2549", "2903", "2963", "2969", "3023", "3329", "3389", "3449", "3491", "3623", "3803" ]

Primes = 3, 9, 11 (mod 20) such that 2p+1 is also prime.

A000356

[ "1", "5", "35", "294", "2772", "28314", "306735", "3476330", "40831076", "493684828", "6114096716", "77266057400", "993420738000", "12964140630900", "171393565105575", "2291968851019650", "30961684478686500", "422056646314726500" ]

Number of rooted cubic maps with 2n nodes and a distinguished Hamiltonian cycle: (2n)!(2n+1)! / (n!^2*(n+1)!(n+2)!).

A000357

[ "1", "1", "5", "35", "315", "3455", "44590", "660665", "11035095", "204904830", "4183174520", "93055783320", "2238954627848", "57903797748386", "1601122732128779", "47120734323344439", "1470076408565099152", "48449426629560437576", "1681560512531504058350", "61293054886119796799892" ]

Number of 5-level labeled rooted trees with n leaves.

A000358

[ "1", "2", "2", "3", "3", "5", "5", "8", "10", "15", "19", "31", "41", "64", "94", "143", "211", "329", "493", "766", "1170", "1811", "2787", "4341", "6713", "10462", "16274", "25415", "39651", "62075", "97109", "152288", "238838", "375167", "589527", "927555", "1459961", "2300348", "3626242", "5721045", "9030451", "14264309", "22542397", "35646312", "56393862", "89264835", "141358275" ]

Number of binary necklaces of length n with no subsequence 00, excluding the necklace "0".

A000359

[ "1", "5", "40", "440", "6170", "105315", "2120610", "49242470", "1296133195", "38152216495", "1242274374380", "44345089721923", "1722416374173854", "72330102999829054", "3265871028909088036", "157797437377747327987", "8124524883679977475839", "444098724261935142753430" ]

Coefficients of iterated exponentials.

A000360

[ "1", "0", "1", "1", "1", "1", "2", "0", "2", "2", "2", "1", "3", "1", "2", "1", "2", "2", "4", "1", "4", "3", "3", "1", "4", "2", "4", "2", "3", "2", "3", "0", "3", "3", "4", "2", "6", "3", "5", "2", "5", "4", "7", "2", "6", "4", "4", "1", "5", "3", "6", "3", "6", "4", "6", "1", "5", "4", "5", "2", "5", "2", "3", "1", "3", "3", "6", "2", "7", "5", "6", "2", "8", "5", "9", "4", "8", "5", "7", "1", "7", "6", "9", "4", "11", "6", "9", "3", "8", "6", "10", "3", "8", "5", "5", "1", "6", "4", "8", "4", "9", "6", "9", "2" ]

Distribution of nonempty triangles inside a fractal rep-4-tile.

A000361

[ "1", "0", "2", "1", "1", "2", "5", "0", "10", "6", "3", "2", "19", "2", "10", "1", "5", "10", "89", "1", "170", "28", "7", "2", "71", "12", "170", "5", "25", "10", "21", "0", "42", "26", "51", "10", "1251", "38", "682", "6", "301", "170", "5833", "3", "2730", "120", "15", "2", "271", "56" ]

From a fractal set of positive Lebesgue measure, a self-replicating tiling with holes, the 4-reptile following the 2-reptile of Paul Levy.

A000362

[ "5", "57", "352", "1280", "3522", "7970", "15872", "29184", "49410", "79042", "122400", "180224", "257314", "362340", "492032", "655360", "867588", "1117314", "1420320", "1803264", "2237380", "2745154", "3380736", "4080640", "4881250", "5874150", "6928416", "8126464", "9600870", "11133604" ]

Generalized class numbers c_(n,2).

A000363

[ "5", "61", "479", "3111", "18270", "101166", "540242", "2819266", "14494859", "73802835", "373398489", "1881341265", "9453340172", "47417364268", "237571096820", "1189405165908", "5951965440609", "29775517732665", "148927275340835", "744793282001995" ]

Number of permutations of [n] with exactly 2 increasing runs of length at least 2.

A000364

[ "1", "1", "5", "61", "1385", "50521", "2702765", "199360981", "19391512145", "2404879675441", "370371188237525", "69348874393137901", "15514534163557086905", "4087072509293123892361", "1252259641403629865468285", "441543893249023104553682821", "177519391579539289436664789665" ]

Euler (or secant or "Zig") numbers: e.g.f. (even powers only) sec(x) = 1/cos(x).

A000365

[ "5", "93", "1030", "8885", "65954", "442610", "2762412", "16322085", "92400330", "505403910", "2687477780", "13957496098", "71053094420", "355548314180", "1752827693528", "8529176056965", "41026491589722", "195327793313790", "921451498774660", "4311086414580022", "20019238138410940" ]

Number of genus 0 rooted planar maps with 4 faces and n vertices.

A000366

[ "1", "1", "2", "7", "38", "295", "3098", "42271", "726734", "15366679", "391888514", "11860602415", "420258768950", "17233254330343", "809698074358250", "43212125903877439", "2599512037272630686", "175079893678534943287", "13122303354155987156306" ]

Genocchi numbers of second kind (A005439) divided by 2^(n-1).

A000367

[ "1", "1", "-1", "1", "-1", "5", "-691", "7", "-3617", "43867", "-174611", "854513", "-236364091", "8553103", "-23749461029", "8615841276005", "-7709321041217", "2577687858367", "-26315271553053477373", "2929993913841559", "-261082718496449122051" ]

Numerators of Bernoulli numbers B_2n.

A000368

[ "1", "1", "4", "9", "28", "71", "202", "542", "1507", "4114", "11381", "31349", "86845", "240567", "668553", "1860361", "5188767", "14495502", "40572216", "113743293", "319405695", "898288484", "2530058013", "7135848125", "20152898513", "56986883801" ]

Number of connected graphs with one cycle of length 4.

A000369

[ "1", "3", "1", "21", "9", "1", "231", "111", "18", "1", "3465", "1785", "345", "30", "1", "65835", "35595", "7650", "825", "45", "1", "1514205", "848925", "196245", "24150", "1680", "63", "1", "40883535", "23586255", "5755050", "775845", "62790", "3066", "84", "1", "1267389585", "748471185", "190482705", "27478710" ]

Triangle of numbers related to triangle A049213; generalization of Stirling numbers of second kind A008277, Bessel triangle A001497.

A000370

[ "1", "2", "4", "14", "222", "616126", "200253952527184", "263735716028826576482466871188128", "5609038300883759793482640992086670939164957990135057216103303119630336" ]

Number of NPN-equivalence classes of Boolean functions of n or fewer variables.

A000371

[ "2", "2", "10", "218", "64594", "4294642034", "18446744047940725978", "340282366920938463334247399005993378250", "115792089237316195423570985008687907850547725730273056332267095982282337798562" ]

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*2^(2^k).

A000372

[ "2", "3", "6", "20", "168", "7581", "7828354", "2414682040998", "56130437228687557907788" ]

Dedekind numbers or Dedekind's problem: number of monotone Boolean functions of n variables, number of antichains of subsets of an n-set, number of elements in a free distributive lattice on n generators, number of Sperner families.

A000373

[ "0", "0", "1", "8", "44", "214", "1000", "4592", "20888", "94846", "434973", "2042836", "9979086", "51460622", "283839957", "1688139424", "10859199656", "75338888918", "560740210491", "4445766353604", "37329808482989", "330143634313064", "3064464030121369" ]

Conjectured dimension of a module associated with the free commutative Moufang loop with n generators.

A000374

[ "1", "1", "2", "1", "2", "2", "3", "1", "3", "2", "2", "2", "2", "3", "5", "1", "3", "3", "2", "2", "6", "2", "3", "2", "3", "2", "4", "3", "2", "5", "7", "1", "5", "3", "6", "3", "2", "2", "5", "2", "3", "6", "4", "2", "8", "3", "3", "2", "5", "3", "8", "2", "2", "4", "5", "3", "5", "2", "2", "5", "2", "7", "13", "1", "7", "5", "2", "3", "6", "6", "3", "3", "9", "2", "8", "2", "6", "5", "3", "2", "5", "3", "2", "6", "12", "4", "5", "2", "9", "8", "10", "3", "14", "3", "5", "2", "3", "5", "8", "3" ]

Number of cycles (mod n) under doubling map.

A000375

[ "0", "1", "2", "4", "7", "10", "16", "22", "30", "38", "51", "65", "80", "101", "113", "139", "159", "191", "221" ]

Topswops (1): start by shuffling n cards labeled 1..n. If top card is m, reverse order of top m cards, then repeat. a(n) is the maximal number of steps before top card is 1.

A000376

[ "0", "1", "2", "4", "7", "10", "16", "22", "30", "38", "51", "63", "80", "101", "112", "130", "159", "191", "207", "231" ]

Topswops (2): start by shuffling n cards labeled 1..n. If top card is m, reverse order of top m cards. Repeat until 1 gets to top, then stop. Suppose the whole deck is now sorted (if not, discard this case). a(n) is the maximal number of steps before 1 got to the top.

A000377

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

Expansion of f(-q^3) * f(-q^8) * chi(-q^12) / chi(-q) in powers of q where chi(), f() are Ramanujan theta functions.

A000378

[ "0", "1", "2", "3", "4", "5", "6", "8", "9", "10", "11", "12", "13", "14", "16", "17", "18", "19", "20", "21", "22", "24", "25", "26", "27", "29", "30", "32", "33", "34", "35", "36", "37", "38", "40", "41", "42", "43", "44", "45", "46", "48", "49", "50", "51", "52", "53", "54", "56", "57", "58", "59", "61", "62", "64", "65", "66", "67", "68", "69", "70", "72", "73", "74", "75", "76", "77", "78", "80", "81", "82", "83" ]

Sums of three squares: numbers of the form x^2 + y^2 + z^2.

A000379

[ "1", "6", "8", "10", "12", "14", "15", "18", "20", "21", "22", "26", "27", "28", "32", "33", "34", "35", "36", "38", "39", "44", "45", "46", "48", "50", "51", "52", "55", "57", "58", "62", "63", "64", "65", "68", "69", "74", "75", "76", "77", "80", "82", "85", "86", "87", "91", "92", "93", "94", "95", "98", "99", "100", "106", "111", "112", "115", "116", "117", "118", "119", "120", "122", "123", "124", "125", "129" ]

Numbers n where total number of 1-bits in the exponents of their prime factorization is even; a 2-way classification of integers: complement of A000028.

A000380

[ "6", "8", "40", "176", "1421", "10352", "93114", "912920", "9929997", "117970704", "1521176826", "21150414880", "315400444070", "5020920314016", "84979755347122", "1523710321272384", "28851091193764023", "575253584489378040", "12047084261153160394", "264377395040950523112", "6066972656940255290199" ]

Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-3 places.

A000381

[ "2", "3", "4", "6", "9", "14", "22", "35", "56", "90", "145", "234", "378", "611", "988", "1598", "2585", "4182", "6766" ]

Essentially the same as A001611.

A000382

[ "6", "11", "20", "36", "65", "119", "218", "400", "735", "1351", "2484", "4568", "8401", "15451", "28418", "52268", "96135", "176819", "325220", "598172", "1100209", "2023599", "3721978", "6845784", "12591359", "23159119", "42596260", "78346736", "144102113", "265045107", "487493954" ]

Restricted permutations.

A000383

[ "1", "1", "1", "1", "1", "1", "6", "11", "21", "41", "81", "161", "321", "636", "1261", "2501", "4961", "9841", "19521", "38721", "76806", "152351", "302201", "599441", "1189041", "2358561", "4678401", "9279996", "18407641", "36513081", "72426721", "143664401", "284970241", "565262081", "1121244166", "2224080691", "4411648301" ]

Hexanacci numbers with a(0) = ... = a(5) = 1.

A000384

[ "0", "1", "6", "15", "28", "45", "66", "91", "120", "153", "190", "231", "276", "325", "378", "435", "496", "561", "630", "703", "780", "861", "946", "1035", "1128", "1225", "1326", "1431", "1540", "1653", "1770", "1891", "2016", "2145", "2278", "2415", "2556", "2701", "2850", "3003", "3160", "3321", "3486", "3655", "3828", "4005", "4186", "4371", "4560" ]

Hexagonal numbers: a(n) = n*(2*n-1).

A000385

[ "1", "6", "17", "38", "70", "116", "185", "258", "384", "490", "686", "826", "1124", "1292", "1705", "1896", "2491", "2670", "3416", "3680", "4602", "4796", "6110", "6178", "7700", "7980", "9684", "9730", "12156", "11920", "14601", "14752", "17514", "17224", "21395", "20406", "24590", "24556", "28920", "27860", "34112", "32186", "38674", "37994", "43980", "42136", "51646", "47772", "56749", "55500", "64316", "60606", "73420", "67956", "80500", "77760", "88860", "83810", "102284", "92690", "108752", "105236", "120777", "112672", "135120", "123046", "145194", "138656", "157512", "146580", "177515", "159396", "185744", "179122" ]

Convolution of A000203 with itself.

A000386

[ "0", "0", "0", "1", "6", "20", "134", "915", "7324", "65784", "657180", "7223637", "86637650", "1125842556", "15757002706", "236298742375", "3780061394232", "64251145312880", "1156374220457784", "21968796934412649", "439337048505773790", "9225384943965382564", "202945418255342821470" ]

Coefficients of ménage hit polynomials.

A000387

[ "0", "0", "1", "0", "6", "20", "135", "924", "7420", "66744", "667485", "7342280", "88107426", "1145396460", "16035550531", "240533257860", "3848532125880", "65425046139824", "1177650830516985", "22375365779822544", "447507315596451070", "9397653627525472260", "206748379805560389951" ]

Rencontres numbers: number of permutations of [n] with exactly two fixed points.

A000388

[ "6", "20", "180", "1106", "9292", "82980", "831545", "9139482", "109595496", "1423490744", "19911182207", "298408841160", "4770598226296", "81037124739588", "1457607971046492", "27675791180024802", "553166885187641670", "11609691036091870428", "255273744004170486155", "5868308906885934514178" ]

Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-2 places.

A000389

[ "0", "0", "0", "0", "0", "1", "6", "21", "56", "126", "252", "462", "792", "1287", "2002", "3003", "4368", "6188", "8568", "11628", "15504", "20349", "26334", "33649", "42504", "53130", "65780", "80730", "98280", "118755", "142506", "169911", "201376", "237336", "278256", "324632", "376992", "435897", "501942", "575757", "658008", "749398" ]

Binomial coefficients C(n,5).

A000390

[ "1", "6", "21", "71", "216", "657", "1907", "5507", "15522", "43352", "119140", "323946", "869476", "2308071", "6056581", "15724170", "40393693", "102736274", "258790004", "645968054", "1598460229", "3923114261", "9554122089", "23098084695", "55458417125", "132293945737", "313657570114" ]

Number of 5-dimensional partitions of n.

A000391

[ "1", "6", "21", "71", "216", "672", "1982", "5817", "16582", "46633", "128704", "350665", "941715", "2499640", "6557378", "17024095", "43756166", "111433472", "281303882", "704320180", "1749727370", "4314842893", "10565857064", "25700414815", "62115621317", "149214574760", "356354881511", "846292135184" ]

Euler transform of A000332.

A000392

[ "0", "0", "0", "1", "6", "25", "90", "301", "966", "3025", "9330", "28501", "86526", "261625", "788970", "2375101", "7141686", "21457825", "64439010", "193448101", "580606446", "1742343625", "5228079450", "15686335501", "47063200806", "141197991025", "423610750290", "1270865805301" ]

Stirling numbers of second kind S(n,3).

A000393

[ "0", "0", "0", "0", "0", "0", "1", "6", "26", "94", "308", "941", "2744", "7722", "21166", "56809", "149971", "390517", "1005491", "2564164", "6485901", "16289602", "40659669", "100934017", "249343899", "613286048", "1502515487", "3667953650", "8925161513", "21652815724", "52387028291" ]

Number of n-node rooted trees of height 6.

A000394

[ "0", "1", "2", "4", "5", "7", "8", "9", "10", "11", "12", "13", "15", "16", "17", "18", "20", "23", "24", "25", "26", "27", "28", "29", "30", "32", "33", "34", "36", "37", "38", "39", "40", "41", "43", "44", "45", "46", "47", "48", "49", "50", "52", "53", "54", "56", "57", "58", "59", "60", "61", "62", "63", "64", "65", "67", "68", "69", "71", "72" ]

Numbers of form x^2 + y^2 + 7z^2.

A000395

[ "1", "6", "27", "104", "369", "1236", "3989", "12522", "38535", "116808", "350064", "1039896", "3068145", "9004182", "26314773", "76652582", "222705603", "645731148", "1869303857", "5404655358", "15611296146", "45060069406", "129989169909", "374843799786", "1080624405287" ]

6th power of rooted tree enumerator; number of linear forests of 6 rooted trees.

A000396

[ "6", "28", "496", "8128", "33550336", "8589869056", "137438691328", "2305843008139952128", "2658455991569831744654692615953842176", "191561942608236107294793378084303638130997321548169216" ]

Perfect numbers k: k is equal to the sum of the proper divisors of k.

A000397

[ "6", "32", "109", "288", "654", "1337", "2506", "4414", "7379", "11822", "18273", "27356", "39938", "56974", "79607", "109267", "147523", "196295", "257715", "334407", "429086", "545034", "685917", "855886", "1059360", "1301776", "1588321", "1925620", "2320544", "2780468", "3314007", "3930001", "4638319", "5449943", "6376505", "7430471", "8625369", "9976540", "11498855", "13210238", "15128487", "17272896", "19664754", "22326319", "25280987", "28554486", "32173404", "36166409", "40563607", "45397395", "50701682", "56512012", "62866699", "69805531", "77370606", "85607286", "94560129", "104280410", "114819255", "126229853", "138570284", "151899428", "166278945", "181775849", "198456941", "216394746", "235661505", "256338017", "278503009", "302242623", "327644632", "354799834", "383805368", "414759214", "447764499", "482931051" ]

Number of partitions into non-integral powers.

A000398

[ "0", "1", "2", "3", "4", "5", "6", "8", "9", "11", "12", "13", "14", "15", "16", "17", "18", "19", "20", "22", "23", "24", "25", "26", "27", "28", "29", "30", "31", "32", "33", "34", "36", "37", "38", "39", "40", "41", "43", "44", "45", "46", "47", "48", "49", "50", "51", "52", "53", "54", "55", "56", "57", "58", "59", "60", "61", "62", "63", "64" ]

Numbers of form x^2 + 2y^2 + 2yz + 4z^2.

A000399

[ "1", "6", "35", "225", "1624", "13132", "118124", "1172700", "12753576", "150917976", "1931559552", "26596717056", "392156797824", "6165817614720", "102992244837120", "1821602444624640", "34012249593822720", "668609730341153280", "13803759753640704000" ]

Unsigned Stirling numbers of first kind s(n,3).

A000400

[ "1", "6", "36", "216", "1296", "7776", "46656", "279936", "1679616", "10077696", "60466176", "362797056", "2176782336", "13060694016", "78364164096", "470184984576", "2821109907456", "16926659444736", "101559956668416", "609359740010496", "3656158440062976", "21936950640377856", "131621703842267136" ]

Powers of 6: a(n) = 6^n.