christopher/oeis · Datasets at Hugging Face

a-number stringlengths 7 7	sequence sequencelengths 1 377	description stringlengths 3 852
A000501	[ "1", "1", "3", "10", "27", "74", "201", "548", "1490", "4051", "11013", "29937", "81377", "221206", "601302", "1634508", "4443055", "12077476", "32829984", "89241150", "242582597", "659407867", "1792456423", "4872401723", "13244561064", "36002449668", "97864804714", "266024120300", "723128532145", "1965667148572" ]	a(n) = floor(cosh(n)).
A000502	[ "42", "1586", "31388", "442610", "5030004", "49145460", "429166584", "3435601554", "25658464260", "181055975100", "1218601601672", "7880146275092", "49238911113224", "298652277299880", "1764885293279472", "10192638073849554", "57674223198273444", "320430129184331628", "1751190732477786600", "9428906326013866076" ]	Number of genus 0 rooted maps with 6 faces and n vertices.
A000503	[ "0", "1", "-3", "-1", "1", "-4", "-1", "0", "-7", "-1", "0", "-226", "-1", "0", "7", "-1", "0", "3", "-2", "0", "2", "-2", "0", "1", "-3", "-1", "1", "-4", "-1", "0", "-7", "-1", "0", "-76", "-1", "0", "7", "-1", "0", "3", "-2", "0", "2", "-2", "0", "1", "-3", "-1", "1", "-4", "-1", "0", "-7", "-1", "0", "-46", "-1", "0", "8", "-1", "0", "3", "-2", "0", "2", "-2", "0", "1", "-3", "-1", "1", "-4", "-1", "0", "-6", "-1", "0", "-33", "-1", "0", "9", "-1", "0", "3", "-2", "0", "2", "-2", "0", "1", "-2", "-1", "1", "-3", "-1", "0", "-6", "-1", "0", "-26" ]	a(n) = floor(tan(n)).
A000504	[ "1", "56", "1918", "56980", "1636635", "47507460", "1422280860", "44346982680", "1446733012725", "49473074851200", "1774073543492250", "66681131440423500", "2624634287988087375", "108060337458000427500", "4647703259223579555000", "208548093035794902390000", "9749651260035434678555625" ]	S2(j,2j+3) where S2(n,k) is a 2-associated Stirling number of the second kind.
A000505	[ "1", "57", "1191", "15619", "156190", "1310354", "9738114", "66318474", "423281535", "2571742175", "15041229521", "85383238549", "473353301060", "2575022097600", "13796160184500", "73008517581444", "382493246941965", "1987497491971605", "10258045633638475" ]	Eulerian numbers (Euler's triangle: column k=5 of A008292, column k=4 of A173018).
A000506	[ "61", "841", "7311", "51663", "325446", "1910706", "10715506", "58258210", "309958755", "1623847695", "8412276585", "43220104041", "220683627988", "1121561317408", "5679711010548", "28683869195556", "144552802373145", "727271783033445" ]	One half of the number of permutations of [n] such that the differences have 5 runs with the same signs.
A000507	[ "61", "1385", "19028", "206276", "1949762", "16889786", "137963364", "1081702420", "8236142455", "61386982075", "450403628440", "3266265481144", "23480284103492", "167687984079924", "1191656966048088", "8436830209386360", "59563995267159825", "419628657826253805" ]	Number of permutations of [n] with exactly 3 increasing runs of length at least 2.
A000508	[ "61", "2763", "38528", "249856", "1066590", "3487246", "9493504", "22634496", "48649086", "96448478", "179369856", "315621376", "530788622", "860061996", "1346126848", "2046820352", "3038120316", "4403100222", "6254596992", "8737505280", "11992903772" ]	Generalized class numbers.
A000509	[ "6", "6", "8", "10", "12", "13", "14", "14", "17", "21", "22", "24" ]	Size of second largest n-arc in PG(2,q), where q runs through the primes and prime powers >= 7.
A000510	[ "7", "9", "9", "11", "15", "15", "17" ]	Maximal number of points in PG(2,q) with at most 3 on a line (next term is 21 or 22).
A000511	[ "1", "1", "2", "3", "5", "8", "11", "17", "25", "33", "47", "67", "87", "117", "160", "207", "270", "356", "455", "584", "751", "945", "1195", "1513", "1882", "2345", "2927", "3608", "4446", "5483", "6701", "8180", "9986", "12109", "14664", "17750", "21371", "25694", "30872", "36937", "44127", "52672", "62658", "74429", "88327", "104524", "123518", "145819", "171737", "201990", "237332", "278289", "325901", "381278", "445272", "519381", "605230", "704170", "818357", "950150", "1101634", "1275907", "1476384", "1706226", "1969869", "2272224", "2618007", "3013559", "3465917", "3982025", "4570898", "5242569", "6007170", "6877474", "7867709", "8992510", "10269905", "11719991", "13363733", "15226469", "17336450", "19723485", "22423058", "25474712", "28920541", "32810028", "37198284", "42144403", "47717124", "53992936", "61054313", "68996364", "77924848", "87954283", "99215750", "111854888" ]	Number of n-step spiral self-avoiding walks on hexagonal lattice, where at each step one may continue in same direction or make turn of 2*Pi/3 counterclockwise.
A000512	[ "0", "0", "1", "1", "2", "7", "16", "51", "224", "1165", "7454", "56349", "481309", "4548786", "46829325", "519812910", "6177695783", "78190425826", "1049510787100", "14886252250208", "222442888670708", "3492326723315796", "57468395960854710", "989052970923320185", "17767732298980160822", "332572885090541084172", "6475438355244504235759", "130954580036269713385884" ]	Number of equivalence classes of n X n matrices over {0,1} with rows and columns summing to 3, where equivalence is defined by row and column permutations.
A000513	[ "0", "0", "0", "1", "1", "4", "16", "194", "3529", "121790", "5582612", "317579783", "21543414506", "1711281449485", "157117486414656", "16502328443493967", "1965612709107379155", "263512349078757245789", "39497131936385398782814", "6579940884199010139737829", "1211896874083479131415289345", "245593008009270037388205883048" ]	Number of equivalence classes of n X n matrices over {0,1} with rows and columns summing to 4, where equivalence is defined by row and column permutations. Also number of isomorphism classes of bicolored quartic bipartite graphs, where isomorphism cannot exchange the colors.
A000514	[ "1", "120", "4293", "88234", "1310354", "15724248", "162512286", "1505621508", "12843262863", "102776998928", "782115518299", "5717291972382", "40457344748072", "278794377854832", "1879708669896492", "12446388300682056", "81180715002105741" ]	Eulerian numbers (Euler's triangle: column k=6 of A008292, column k=5 of A173018)
A000515	[ "1", "12", "180", "2800", "44100", "698544", "11099088", "176679360", "2815827300", "44914183600", "716830370256", "11445589052352", "182811491808400", "2920656969720000", "46670906271240000", "745904795339462400", "11922821963004219300", "190600129650794094000", "3047248986392325330000" ]	a(n) = (2n)!(2n+1)!/n!^4, or equally (2n+1)*binomial(2n,n)^2.
A000516	[ "0", "0", "0", "0", "1", "1", "4", "51", "3529", "601055", "156473848", "54062069505", "23869437984682", "13186966476208771", "8971034249976338907", "7414924597575224629299", "7360058177440420943520750", "8683626883245180573511018830", "12066478410398147578519948851818", "19585444567548740264243478805318202" ]	Number of equivalence classes of n X n matrices over {0,1} with rows and columns summing to 5, where equivalence is defined by row and column permutations. Isomorphism classes of bicolored 5-regular bipartite graphs, where isomorphism cannot exchange the colors.
A000517	[ "272", "7936", "137216", "1841152", "21253376", "222398464", "2174832640", "20261765120", "182172651520", "1594922762240", "13684856848384", "115620218667008", "965271355195392", "7984436548730880", "65569731961159680", "535438370914959360", "4353038473793372160", "35266789418949672960" ]	Number of permutations of length n with exactly three valleys.
A000518	[ "272", "24611", "515086", "4456448", "23750912", "93241002", "296327464", "806453248", "1951153920", "4300685074", "8787223186", "16878338048", "30768878848", "53624926972", "89982082488", "146028888064", "230022888960", "353194774434", "529896144586" ]	Generalized tangent numbers d_(n,4).
A000519	[ "1", "2", "3", "5", "7", "18", "43", "313", "7525", "846992", "324127859", "403254094631", "1555631972009429", "19731915624463099552", "791773335030637885025287", "107432353216118868234728540267", "47049030539260648478475949282317451", "71364337698829887974206671525372672234854" ]	Number of equivalence classes of nonzero regular 0-1 matrices of order n.
A000520	[ "0", "0", "0", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2" ]	Nearest integer to log_10(n).
A000521	[ "1", "744", "196884", "21493760", "864299970", "20245856256", "333202640600", "4252023300096", "44656994071935", "401490886656000", "3176440229784420", "22567393309593600", "146211911499519294", "874313719685775360", "4872010111798142520", "25497827389410525184", "126142916465781843075" ]	Coefficients of modular function j as power series in q = e^(2 Pi i t). Another name is the elliptic modular invariant J(tau).
A000522	[ "1", "2", "5", "16", "65", "326", "1957", "13700", "109601", "986410", "9864101", "108505112", "1302061345", "16926797486", "236975164805", "3554627472076", "56874039553217", "966858672404690", "17403456103284421", "330665665962404000", "6613313319248080001", "138879579704209680022", "3055350753492612960485", "70273067330330098091156" ]	Total number of ordered k-tuples (k=0..n) of distinct elements from an n-element set: a(n) = Sum_{k=0..n} n!/k!.
A000523	[ "0", "1", "1", "2", "2", "2", "2", "3", "3", "3", "3", "3", "3", "3", "3", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "5", "5", "5", "5", "5", "5", "5", "5", "5", "5", "5", "5", "5", "5", "5", "5", "5", "5", "5", "5", "5", "5", "5", "5", "5", "5", "5", "5", "5", "5", "5", "5", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6" ]	a(n) = floor(log_2(n)).
A000524	[ "2", "9", "34", "119", "401", "1316", "4247", "13532", "42712", "133816", "416770", "1291731", "3987444", "12266845", "37627230", "115125955", "351467506", "1070908135", "3257389088", "9892759091", "30002923380", "90879555521", "274963755791", "831064788976" ]	Number of rooted trees with n nodes, 2 of which are labeled.
A000525	[ "64", "625", "4016", "21256", "100407", "439646", "1823298", "7258228", "27983518", "105146732", "386812476", "1398023732", "4977320988", "17492710572", "60790051789", "209179971147", "713533304668", "2415061934763", "8117293752058", "27111950991825", "90039381031273" ]	Number of partially labeled rooted trees with n nodes (4 of which are labeled).
A000526	[ "125", "1296", "8716", "47787", "232154", "1040014", "4395772", "17781210", "69498964", "264248924", "982218072", "3582421612", "12857819052", "45515994861", "159205157535", "551049504784", "1889714853263", "6427147635062", "21698583468717" ]	Number of partially labeled trees with n nodes (5 of which are labeled).
A000527	[ "52", "472", "3224", "18888", "101340", "511120", "2465904", "11496144", "52165892", "231557064", "1009247192", "4331502840", "18346242492", "76822836544", "318485778848", "1308750158016", "5335993098340", "21603437175288", "86912657626392", "347660876627944", "1383457374046444", "5479086968052912", "21604984733546336", "84850331177724944", "332001521469767940", "1294589169323791912", "5031934808360234760", "19500424806065865400", "75360646947991208396", "290478417300879735680", "1116919455364101145920", "4284817000807140094464", "16402243457215852326116", "62659647762404302956856", "238910441445219175239480" ]	Series-parallel numbers.
A000528	[ "1", "1", "1", "2", "2", "17", "324", "842227", "57810418543", "104452188344901572", "6108088657705958932053657" ]	Number of types of Latin squares of order n. Equivalently, number of nonisomorphic 1-factorizations of K_{n,n}.
A000529	[ "20", "74", "186", "388", "721", "1236", "1995", "3072", "4554", "6542", "9152", "12516", "16783", "22120", "28713", "36768", "46512", "58194", "72086", "88484", "107709", "130108", "156055", "185952", "220230", "259350", "303804", "354116", "410843", "474576", "545941", "625600", "714252", "812634", "921522", "1041732", "1174121", "1319588", "1479075", "1653568", "1844098", "2051742", "2277624", "2522916", "2788839", "3076664", "3387713", "3723360", "4085032", "4474210" ]	Powers of rooted tree enumerator.
A000530	[ "1", "5", "28", "226", "2077", "20770", "222884", "2529541" ]	Let p(n, s, x) be predicate that number of occurrences of s's in x >= 2n - the length of the longest sequence of s's in x. Then a(n)=#{x in {0,1} \| x ends in 0 and p(n,0,x) and (there is no prefix y of x such that p(n,0,y) or p(n,1,y))}.
A000531	[ "1", "7", "38", "187", "874", "3958", "17548", "76627", "330818", "1415650", "6015316", "25413342", "106853668", "447472972", "1867450648", "7770342787", "32248174258", "133530264682", "551793690628", "2276098026922", "9373521044908", "38546133661492" ]	From area of cyclic polygon of 2n + 1 sides.
A000532	[ "1", "1", "2", "8", "86", "1770", "88418", "8934966", "2087813834", "1013346943033", "1111598871478668", "2568944901392936854", "13251059359839620127088", "145194816279817259193401518", "3524171261632305641165676374930" ]	Number of Hamiltonian paths from NW to SW corners in an n X n grid.
A000533	[ "1", "11", "101", "1001", "10001", "100001", "1000001", "10000001", "100000001", "1000000001", "10000000001", "100000000001", "1000000000001", "10000000000001", "100000000000001", "1000000000000001", "10000000000000001" ]	a(0)=1; a(n) = 10^n + 1, n >= 1.
A000534	[ "0", "1", "2", "3", "5", "6", "8", "9", "11", "14", "17", "24", "29", "32", "41", "56", "96", "128", "224", "384", "512", "896", "1536", "2048", "3584", "6144", "8192", "14336", "24576", "32768", "57344", "98304", "131072", "229376", "393216", "524288", "917504", "1572864", "2097152", "3670016", "6291456", "8388608", "14680064" ]	Numbers that are not the sum of 4 nonzero squares.
A000535	[ "0", "27", "378", "4536", "48600", "489780", "4738104", "44535456", "409752432", "3708359550", "33125746500", "292779558720", "2565087894720", "22307854940280", "192788833482000", "1657111548654720", "14176605442521312", "120779466450505758", "1025230099571720676", "8674221270307971600" ]	Card matching: coefficients B[n,2] of t^2 in the reduced hit polynomial A[n,n,n](t).
A000536	[ "24", "240", "2520", "26880", "304080", "3671136", "47391120", "653463360", "9603708840", "150046937040", "2485510331304", "43536519673920", "804343214307360", "15636586027419840", "319143375070100640", "6824486562845878656", "152599994618389811640", "3561710724832153990320", "86627571138529803385080", "2192153071078356814538880", "57633178354598014299807984", "1572073330365520093029415200", "44434609885866805678475703600", "1299879247128621094998213278400", "39312834919322919649653205283400", "1227895179113516869799082638629776", "39569125440836907870479047149487560", "1314368274045259508166257769617810880", "44963797526832537006635800892057862720", "1582832153412276057834241761650127323520" ]	Number of 3-line Latin rectangles.
A000537	[ "0", "1", "9", "36", "100", "225", "441", "784", "1296", "2025", "3025", "4356", "6084", "8281", "11025", "14400", "18496", "23409", "29241", "36100", "44100", "53361", "64009", "76176", "90000", "105625", "123201", "142884", "164836", "189225", "216225", "246016", "278784", "314721", "354025", "396900", "443556", "494209", "549081" ]	Sum of first n cubes; or n-th triangular number squared.
A000538	[ "0", "1", "17", "98", "354", "979", "2275", "4676", "8772", "15333", "25333", "39974", "60710", "89271", "127687", "178312", "243848", "327369", "432345", "562666", "722666", "917147", "1151403", "1431244", "1763020", "2153645", "2610621", "3142062", "3756718", "4463999", "5273999", "6197520", "7246096", "8432017", "9768353" ]	Sum of fourth powers: 0^4 + 1^4 + ... + n^4.
A000539	[ "0", "1", "33", "276", "1300", "4425", "12201", "29008", "61776", "120825", "220825", "381876", "630708", "1002001", "1539825", "2299200", "3347776", "4767633", "6657201", "9133300", "12333300", "16417401", "21571033", "28007376", "35970000", "45735625", "57617001", "71965908", "89176276", "109687425", "133987425" ]	Sum of 5th powers: 0^5 + 1^5 + 2^5 + ... + n^5.
A000540	[ "0", "1", "65", "794", "4890", "20515", "67171", "184820", "446964", "978405", "1978405", "3749966", "6735950", "11562759", "19092295", "30482920", "47260136", "71397705", "105409929", "152455810", "216455810", "302221931", "415601835", "563637724", "754740700", "998881325", "1307797101", "1695217590" ]	Sum of 6th powers: 0^6 + 1^6 + 2^6 + ... + n^6.
A000541	[ "0", "1", "129", "2316", "18700", "96825", "376761", "1200304", "3297456", "8080425", "18080425", "37567596", "73399404", "136147921", "241561425", "412420800", "680856256", "1091194929", "1703414961", "2597286700", "3877286700", "5678375241", "8172733129", "11577558576", "16164030000", "22267545625" ]	Sum of 7th powers: 1^7 + 2^7 + ... + n^7.
A000542	[ "0", "1", "257", "6818", "72354", "462979", "2142595", "7907396", "24684612", "67731333", "167731333", "382090214", "812071910", "1627802631", "3103591687", "5666482312", "9961449608", "16937207049", "27957167625", "44940730666", "70540730666", "108363590027", "163239463563", "241550448844" ]	Sum of 8th powers: 1^8 + 2^8 + ... + n^8.
A000543	[ "0", "1", "23", "333", "2916", "16725", "70911", "241913", "701968", "1798281", "4173775", "8942021", "17930628", "34009053", "61518471", "106823025", "179003456", "290715793", "459239463", "707740861", "1066780100", "1576090341", "2286660783", "3263156073", "4586706576" ]	Number of inequivalent ways to color vertices of a cube using at most n colors.
A000544	[ "3", "25", "155", "1005", "7488", "64164", "619986", "6646750", "78161249", "999473835", "13801761213", "204631472475", "3241541125110", "54629642149630", "975867376041308", "18416844056075364", "366128842105397631", "7647337600268371485", "167424323805645018159", "3833790834030516355705", "91641405910147125954428", "2282611988081527293910920" ]	Number of permutations of length n by rises.
A000545	[ "1", "96", "9099", "280832", "4073375", "36292320", "230719293", "1145393152", "4707296613", "16666924000", "52307593239", "148602435840", "388302646355", "944900450144", "2162441849625", "4691253854208", "9710376716137", "19280531603808", "36888593841475", "68266682784000", "122597146773927" ]	Number of ways of n-coloring a dodecahedron.
A000546	[ "1", "12", "28", "314", "98", "386", "943", "1494", "1680", "4722", "6576", "11696", "3982", "2987", "17548", "36208", "7083", "59692", "159116", "79592", "57857", "212160", "352258", "221185", "57346", "294913", "252548", "530052", "331778", "524289", "1088129", "913319", "2065786", "1541308", "1032875", "1264924", "8151894" ]	First occurrence of n consecutive numbers that take same number of steps to reach 1 in 3x+1 problem.
A000547	[ "0", "9", "18", "37", "25", "120", "36", "47", "42", "59", "137", "143", "51", "48", "141", "41", "57", "73", "77", "76", "166", "80", "104", "93", "78", "96", "181", "102", "91", "102", "209", "201", "197", "194", "196", "129", "230", "115", "200", "97", "213", "206", "115", "216", "220", "214", "208", "246", "211", "233", "199", "208", "209", "224", "207", "228" ]	Number of steps to reach 1 in sequence A000546.
A000548	[ "1", "4", "9", "16", "36", "49", "64", "81", "121", "144", "196", "256", "324", "361", "441", "484", "529", "576", "729", "784", "961", "1024", "1089", "1296", "1444", "1764", "1849", "1936", "2116", "2209", "2304", "2401", "2916", "3136", "3249", "3481", "3844", "3969", "4096", "4356", "4489" ]	Squares that are not the sum of 2 nonzero squares.
A000549	[ "1", "2", "4", "5", "8", "10", "13", "16", "20", "25", "32", "37", "40", "52", "58", "64", "80", "85", "100", "128", "130", "148", "160", "208", "232", "256", "320", "340", "400", "512", "520", "592", "640", "832", "928", "1024", "1280", "1360", "1600", "2048", "2080", "2368", "2560", "3328", "3712", "4096", "5120", "5440", "6400", "8192", "8320", "9472", "10240" ]	Numbers that are the sum of 2 squares but not sum of 3 nonzero squares.
A000550	[ "1", "3", "14", "42", "128", "334", "850", "2010", "4625", "10201", "21990", "46108", "94912", "191562", "380933", "746338", "1444676", "2763931", "5235309", "9822686", "18275648", "33734658", "61826344", "112550305", "203627610", "366267931", "655261559", "1166312530", "2066048261", "3643352362", "6397485909", "11188129665", "19491131627", "33831897511", "58519577756", "100885389220", "173368983090", "297021470421", "507378371670", "864277569606", "1468245046383", "2487774321958", "4204663810414", "7089200255686", "11924621337321", "20012746962064", "33513139512868", "56001473574091", "93387290773141", "155419866337746" ]	Number of trees of diameter 7.
A000551	[ "6", "36", "200", "1170", "7392", "50568", "372528", "2936070", "24617120", "218521116", "2045278248", "20112821274", "207162957120", "2228888801040", "24989309310944", "291322555295886", "3524580202643136", "44176839081266340", "572725044269255640" ]	Number of labeled rooted trees of height 2 with n nodes.
A000552	[ "24", "300", "3360", "38850", "475776", "6231960", "87530400", "1316954430", "21173760960", "362670636900", "6596214691248", "126980000240730", "2579214238608000", "55118036257959600", "1235935135837111104", "29009023670878484598" ]	Number of labeled rooted trees of height 3 with n nodes.
A000553	[ "120", "2520", "43680", "757680", "13747104", "264181680", "5395040640", "117080049240", "2696387899920", "65774992411128", "1695845836077120", "46110625382246880", "1319345179723609920", "39640903618873667040", "1248193457738661143808" ]	Number of labeled rooted trees of height 4 with n nodes.
A000554	[ "12", "60", "210", "630", "1736", "4536", "11430", "28050", "67452", "159588", "372554", "859950", "1965840", "4456176", "10026702", "22412970", "49806980", "110100060", "242220594", "530578950", "1157627352", "2516581800", "5452594550", "11777604930", "25367149836", "54492396756", "116769422490", "249644973150" ]	Number of labeled trees of diameter 3 with n nodes.
A000555	[ "60", "720", "6090", "47040", "363384", "2913120", "24560910", "218386080", "2044958916", "20112075984", "207161237010", "2228884869120", "24989300398320", "291322535242176", "3524580157816854", "44176838981652000", "572725044049055100", "7668896804089696560", "105920137922879314650", "1507138839384235136640", "22068265782102952223400", "332178010291171425732000", "5135009134117954527323550", "81449458937043220255508640" ]	Number of labeled trees of diameter 4 with n nodes.
A000556	[ "1", "1", "5", "31", "257", "2671", "33305", "484471", "8054177", "150635551", "3130337705", "71556251911", "1784401334897", "48205833997231", "1402462784186105", "43716593539939351", "1453550100421124417", "51350258701767067711", "1920785418183176050505", "75839622064482770570791" ]	Expansion of exp(-x) / (1 - exp(x) + exp(-x)).
A000557	[ "1", "2", "8", "50", "416", "4322", "53888", "783890", "13031936", "243733442", "5064992768", "115780447730", "2887222009856", "77998677862562", "2269232452763648", "70734934220015570", "2351893466832306176", "83086463910558199682", "3107896091715557654528", "122711086194279627711410" ]	Expansion of e.g.f.: 1/(1-2*sinh(x)).
A000558	[ "1", "6", "32", "175", "1012", "6230", "40819", "283944", "2090424", "16235417", "132609666", "1135846062", "10175352709", "95108406130", "925496853980", "9357279554071", "98118527430960", "1065259283215810", "11956366813630835", "138539436100687988", "1655071323662574756", "20361556640795422729" ]	Generalized Stirling numbers of second kind.
A000559	[ "1", "12", "110", "945", "8092", "70756", "638423", "5971350", "57996774", "585092607", "6128147610", "66579524648", "749542556193", "8733648533696", "105203108066962", "1308549777461505", "16787682400875456", "221901108871482760", "3018891886411332135", "42230736603244134242" ]	Generalized Stirling numbers of second kind.
A000560	[ "1", "2", "5", "12", "33", "87", "252", "703", "2105", "6099", "18689", "55639", "173423", "526937", "1664094", "5137233", "16393315", "51255709", "164951529", "521138861", "1688959630", "5382512216", "17547919924", "56335234064", "184596351277", "596362337295", "1962723402375" ]	Number of ways of folding a strip of n labeled stamps.
A000561	[ "6", "44", "145", "336", "644", "1096", "1719", "2540", "3586", "4884", "6461", "8344", "10560", "13136", "16099", "19476", "23294", "27580", "32361", "37664", "43516", "49944", "56975", "64636", "72954", "81956", "91669", "102120", "113336", "125344", "138171", "151844", "166390", "181836", "198209", "215536", "233844", "253160", "273511", "294924", "317426", "341044" ]	Number of discordant permutations.
A000562	[ "9", "95", "420", "1225", "2834", "5652", "10165", "16940", "26625", "39949", "57722", "80835", "110260", "147050", "192339", "247342", "313355", "391755", "484000", "591629", "716262", "859600", "1023425", "1209600", "1420069", "1656857", "1922070", "2217895", "2546600", "2910534", "3312127", "3753890", "4238415", "4768375", "5346524", "5975697" ]	Number of discordant permutations.
A000563	[ "13", "192", "1085", "3880", "10656", "24626", "50380", "94128", "163943", "270004", "424839", "643568", "944146", "1347606", "1878302", "2564152", "3436881", "4532264", "5890369", "7555800", "9577940", "12011194", "14915232", "18355232", "22402123", "27132828", "32630507", "38984800", "46292070" ]	Number of discordant permutations.
A000564	[ "20", "371", "2588", "11097", "35645", "94457", "218124", "454220", "872648", "1571715", "2684936", "4388567", "6909867", "10536089", "15624200", "22611330", "32025950", "44499779", "60780420", "81744725", "108412889", "141963273", "183747956", "235309016", "298395540" ]	Number of discordant permutations.
A000565	[ "31", "696", "5823", "29380", "108933", "327840", "848380", "1958004", "4130895", "8107024", "14990889", "26372124", "44470165", "72305160", "113897310", "174496828", "260846703", "381480456", "547057075", "770735316", "1068589557", "1460069392", "1968505152", "2621661540" ]	Number of discordant permutations.
A000566	[ "0", "1", "7", "18", "34", "55", "81", "112", "148", "189", "235", "286", "342", "403", "469", "540", "616", "697", "783", "874", "970", "1071", "1177", "1288", "1404", "1525", "1651", "1782", "1918", "2059", "2205", "2356", "2512", "2673", "2839", "3010", "3186", "3367", "3553", "3744", "3940", "4141", "4347", "4558", "4774", "4995", "5221", "5452", "5688" ]	Heptagonal numbers (or 7-gonal numbers): n(5n-3)/2.
A000567	[ "0", "1", "8", "21", "40", "65", "96", "133", "176", "225", "280", "341", "408", "481", "560", "645", "736", "833", "936", "1045", "1160", "1281", "1408", "1541", "1680", "1825", "1976", "2133", "2296", "2465", "2640", "2821", "3008", "3201", "3400", "3605", "3816", "4033", "4256", "4485", "4720", "4961", "5208", "5461" ]	Octagonal numbers: n(3n-2). Also called star numbers.
A000568	[ "1", "1", "1", "2", "4", "12", "56", "456", "6880", "191536", "9733056", "903753248", "154108311168", "48542114686912", "28401423719122304", "31021002160355166848", "63530415842308265100288", "244912778438520759443245824", "1783398846284777975419600287232", "24605641171260376770598003978281472" ]	Number of outcomes of unlabeled n-team round-robin tournaments.
A000569	[ "1", "2", "5", "9", "17", "31", "54", "90", "151", "244", "387", "607", "933", "1420", "2136", "3173", "4657", "6799", "9803", "14048", "19956", "28179", "39467", "54996", "76104", "104802", "143481", "195485", "264941", "357635", "480408", "642723", "856398", "1136715", "1503172", "1980785" ]	Number of graphical partitions of 2n.
A000570	[ "1", "1", "2", "4", "7", "11", "18", "31", "53", "89", "149", "251", "424", "715", "1204", "2028", "3418", "5761", "9708", "16358", "27565", "46452", "78279", "131910", "222285", "374581", "631222", "1063696", "1792472", "3020560", "5090059", "8577449", "14454177", "24357268", "41045336", "69167021", "116555915" ]	Number of tournaments on n nodes determined by their score vectors.
A000571	[ "1", "1", "1", "2", "4", "9", "22", "59", "167", "490", "1486", "4639", "14805", "48107", "158808", "531469", "1799659", "6157068", "21258104", "73996100", "259451116", "915695102", "3251073303", "11605141649", "41631194766", "150021775417", "542875459724", "1972050156181", "7189259574618", "26295934251565", "96478910768821", "354998461378719", "1309755903513481" ]	Number of different score sequences that are possible in an n-team round-robin tournament.
A000572	[ "3", "7", "11", "14", "18", "22", "26", "29", "33", "37", "40", "44", "48", "52", "55", "59", "63", "66", "70", "74", "78", "81", "85", "89", "92", "96", "100", "104", "107", "111", "115", "118", "122", "126", "130", "133", "137", "141", "145", "148", "152", "156", "159", "163", "167", "171", "174", "178", "182", "185", "189", "193", "197", "200", "204", "208", "211", "215", "219" ]	A Beatty sequence: [ n(e+1) ].
A000573	[ "4", "56", "6552", "1293216", "420909504", "207624560256", "147174521059584", "143968880078466048", "188237563987982390784", "320510030393570671051776", "695457005987768649183581184", "1888143905499961681708381310976", "6314083806394358817244705266941952", "25655084790196439186603345691314159616" ]	Number of 4 X n normalized Latin rectangles.
A000574	[ "3", "16", "51", "126", "266", "504", "882", "1452", "2277", "3432", "5005", "7098", "9828", "13328", "17748", "23256", "30039", "38304", "48279", "60214", "74382", "91080", "110630", "133380", "159705", "190008", "224721", "264306", "309256", "360096", "417384", "481712", "553707", "634032", "723387", "822510" ]	Coefficient of x^5 in expansion of (1 + x + x^2)^n.
A000575	[ "10", "80", "365", "1246", "3535", "8800", "19855", "41470", "81367", "151580", "270270", "464100", "771290", "1245488", "1960610", "3016820", "4547840", "6729800", "9791859", "14028850", "19816225", "27627600", "38055225", "51833730", "69867525", "93262260", "123360780", "161784040", "210477476", "271763360" ]	Tenth column of quintinomial coefficients.
A000576	[ "1", "3", "46", "6552", "11270400", "335390189568", "224382967916691456", "4292039421591854273003520", "2905990310033882693113989027594240" ]	a(n) is the number of (n-2) X n normalized Latin rectangles.
A000577	[ "1", "1", "1", "3", "4", "12", "24", "66", "160", "448", "1186", "3334", "9235", "26166", "73983", "211297", "604107", "1736328", "5000593", "14448984", "41835738", "121419260", "353045291", "1028452717", "3000800627", "8769216722", "25661961898", "75195166667", "220605519559", "647943626796" ]	Number of triangular polyominoes (or triangular polyforms, or polyiamonds) with n cells (turning over is allowed, holes are allowed, must be connected along edges).
A000578	[ "0", "1", "8", "27", "64", "125", "216", "343", "512", "729", "1000", "1331", "1728", "2197", "2744", "3375", "4096", "4913", "5832", "6859", "8000", "9261", "10648", "12167", "13824", "15625", "17576", "19683", "21952", "24389", "27000", "29791", "32768", "35937", "39304", "42875", "46656", "50653", "54872", "59319", "64000" ]	The cubes: a(n) = n^3.
A000579	[ "0", "0", "0", "0", "0", "0", "1", "7", "28", "84", "210", "462", "924", "1716", "3003", "5005", "8008", "12376", "18564", "27132", "38760", "54264", "74613", "100947", "134596", "177100", "230230", "296010", "376740", "475020", "593775", "736281", "906192", "1107568", "1344904", "1623160", "1947792", "2324784", "2760681", "3262623" ]	Figurate numbers or binomial coefficients C(n,6).
A000580	[ "1", "8", "36", "120", "330", "792", "1716", "3432", "6435", "11440", "19448", "31824", "50388", "77520", "116280", "170544", "245157", "346104", "480700", "657800", "888030", "1184040", "1560780", "2035800", "2629575", "3365856", "4272048", "5379616", "6724520", "8347680", "10295472" ]	a(n) = binomial coefficient C(n,7).
A000581	[ "1", "9", "45", "165", "495", "1287", "3003", "6435", "12870", "24310", "43758", "75582", "125970", "203490", "319770", "490314", "735471", "1081575", "1562275", "2220075", "3108105", "4292145", "5852925", "7888725", "10518300", "13884156", "18156204", "23535820", "30260340", "38608020", "48903492", "61523748", "76904685" ]	a(n) = binomial coefficient C(n,8).
A000582	[ "1", "10", "55", "220", "715", "2002", "5005", "11440", "24310", "48620", "92378", "167960", "293930", "497420", "817190", "1307504", "2042975", "3124550", "4686825", "6906900", "10015005", "14307150", "20160075", "28048800", "38567100", "52451256", "70607460", "94143280", "124403620", "163011640", "211915132" ]	a(n) = binomial coefficient C(n,9).
A000583	[ "0", "1", "16", "81", "256", "625", "1296", "2401", "4096", "6561", "10000", "14641", "20736", "28561", "38416", "50625", "65536", "83521", "104976", "130321", "160000", "194481", "234256", "279841", "331776", "390625", "456976", "531441", "614656", "707281", "810000", "923521", "1048576", "1185921" ]	Fourth powers: a(n) = n^4.
A000584	[ "0", "1", "32", "243", "1024", "3125", "7776", "16807", "32768", "59049", "100000", "161051", "248832", "371293", "537824", "759375", "1048576", "1419857", "1889568", "2476099", "3200000", "4084101", "5153632", "6436343", "7962624", "9765625", "11881376", "14348907", "17210368", "20511149" ]	Fifth powers: a(n) = n^5.
A000585	[ "4", "8", "20", "92", "2744", "950998216", "2076795963681989019155896", "21651217007530946175606768762255421159692845640522169779616" ]	Number of equivalence classes of Boolean functions of n variables under GL(n,2).
A000586	[ "1", "0", "1", "1", "0", "2", "0", "2", "1", "1", "2", "1", "2", "2", "2", "2", "3", "2", "4", "3", "4", "4", "4", "5", "5", "5", "6", "5", "6", "7", "6", "9", "7", "9", "9", "9", "11", "11", "11", "13", "12", "14", "15", "15", "17", "16", "18", "19", "20", "21", "23", "22", "25", "26", "27", "30", "29", "32", "32", "35", "37", "39", "40", "42", "44", "45", "50", "50", "53", "55", "57", "61", "64", "67", "70", "71", "76", "78", "83", "87", "89", "93", "96" ]	Number of partitions of n into distinct primes.
A000587	[ "1", "-1", "0", "1", "1", "-2", "-9", "-9", "50", "267", "413", "-2180", "-17731", "-50533", "110176", "1966797", "9938669", "8638718", "-278475061", "-2540956509", "-9816860358", "27172288399", "725503033401", "5592543175252", "15823587507881", "-168392610536153", "-2848115497132448", "-20819319685262839" ]	Rao Uppuluri-Carpenter numbers (or complementary Bell numbers): e.g.f. = exp(1 - exp(x)).
A000588	[ "0", "0", "0", "1", "7", "35", "154", "637", "2548", "9996", "38760", "149226", "572033", "2187185", "8351070", "31865925", "121580760", "463991880", "1771605360", "6768687870", "25880277150", "99035193894", "379300783092", "1453986335186", "5578559816632", "21422369201800", "82336410323440", "316729578421620" ]	a(n) = 7*binomial(2n,n-3)/(n+4).
A000589	[ "1", "11", "77", "440", "2244", "10659", "48279", "211508", "904475", "3798795", "15737865", "64512240", "262256280", "1059111900", "4254603804", "17018415216", "67837293986", "269638992062", "1069258071970", "4232010895376", "16723268860760", "65997186039785", "260170725132045", "1024713341952300" ]	a(n) = 11*binomial(2n,n-5)/(n+6).
A000590	[ "1", "13", "104", "663", "3705", "19019", "92092", "427570", "1924065", "8454225", "36463440", "154969620", "650872404", "2707475148", "11173706960", "45812198536", "186803188858", "758201178306", "3065415516592", "12352414499425", "49634247352235", "198954083924505", "795816335698020", "3177498557750790" ]	a(n) = 13*binomial(2n,n-6)/(n+7).
A000591	[ "10", "378", "16576", "819470", "45660051", "2846339383", "196946930215", "15006717613499", "1250005718758059", "113076157328915784", "11044120989736000167", "1158658706030435109195", "129976520576914828292552" ]	Number of n-state 2-input 1-output automata with one initial and one terminal state.
A000592	[ "1", "3", "4", "6", "8", "9", "11", "13", "15", "17", "19", "20", "22", "26", "28", "30", "31", "33", "35", "37", "39", "41", "43", "45", "48", "50", "52", "54", "56", "58", "62", "64", "65", "67", "69", "71", "73", "75", "79", "81", "83", "85", "86", "90", "92", "94", "96", "98", "100", "102", "104", "106", "108", "112", "113", "117", "119", "121", "123", "127", "129", "131", "133", "135", "137" ]	Number of nonnegative solutions of x^2 + y^2 = z in first n shells.
A000593	[ "1", "1", "4", "1", "6", "4", "8", "1", "13", "6", "12", "4", "14", "8", "24", "1", "18", "13", "20", "6", "32", "12", "24", "4", "31", "14", "40", "8", "30", "24", "32", "1", "48", "18", "48", "13", "38", "20", "56", "6", "42", "32", "44", "12", "78", "24", "48", "4", "57", "31", "72", "14", "54", "40", "72", "8", "80", "30", "60", "24", "62", "32", "104", "1", "84", "48", "68", "18", "96", "48", "72", "13", "74", "38", "124" ]	Sum of odd divisors of n.
A000594	[ "1", "-24", "252", "-1472", "4830", "-6048", "-16744", "84480", "-113643", "-115920", "534612", "-370944", "-577738", "401856", "1217160", "987136", "-6905934", "2727432", "10661420", "-7109760", "-4219488", "-12830688", "18643272", "21288960", "-25499225", "13865712", "-73279080", "24647168" ]	Ramanujan's tau function (or Ramanujan numbers, or tau numbers).
A000595	[ "1", "2", "10", "104", "3044", "291968", "96928992", "112282908928", "458297100061728", "6666621572153927936", "349390545493499839161856", "66603421985078180758538636288", "46557456482586989066031126651104256", "120168591267113007604119117625289606148096", "1152050155760474157553893461743236772303142428672" ]	Number of binary relations on n unlabeled points.
A000596	[ "4", "49", "273", "1023", "3003", "7462", "16422", "32946", "61446", "108031", "180895", "290745", "451269", "679644", "997084", "1429428", "2007768", "2769117", "3757117", "5022787", "6625311", "8632866", "11123490", "14185990", "17920890", "22441419", "27874539", "34362013", "42061513", "51147768", "61813752", "74271912" ]	Central factorial numbers.
A000597	[ "36", "820", "7645", "44473", "191620", "669188", "1999370", "5293970", "12728936", "28285400", "58856655", "115842675", "217378200", "391367064", "679524340", "1142659012", "1867463260", "2975110060", "4631998657", "7063027565", "10567817084", "15540347900", "22492529150", "32082258390", "45146587200" ]	Central factorial numbers.
A000598	[ "1", "1", "1", "2", "4", "8", "17", "39", "89", "211", "507", "1238", "3057", "7639", "19241", "48865", "124906", "321198", "830219", "2156010", "5622109", "14715813", "38649152", "101821927", "269010485", "712566567", "1891993344", "5034704828", "13425117806", "35866550869", "95991365288", "257332864506", "690928354105" ]	Number of rooted ternary trees with n nodes; number of n-carbon alkyl radicals C(n)H(2n+1) ignoring stereoisomers.
A000599	[ "0", "0", "1", "1", "3", "6", "15", "33", "82", "194", "482", "1188", "2988", "7528", "19181", "49060", "126369", "326863", "849650", "2216862", "5806256", "15256265", "40210657", "106273050", "281593237", "747890675", "1990689459", "5309397294", "14187485959", "37977600390", "101827024251" ]	Number of secondary alcohols (alkanols or alkyl alcohols C_n H_{2n+1} OH) with n carbon atoms.
A000600	[ "0", "0", "0", "0", "1", "1", "3", "7", "17", "40", "102", "249", "631", "1594", "4074", "10443", "26981", "69923", "182158", "476141", "1249237", "3287448", "8677074", "22962118", "60915508", "161962845", "431536102", "1152022025", "3081015684", "8253947104", "22147214029", "59514474967" ]	Number of tertiary alcohols (alkanols or alkyl alcohols C_n H_{2n+1} OH) with n carbon atoms.

A000501

[ "1", "1", "3", "10", "27", "74", "201", "548", "1490", "4051", "11013", "29937", "81377", "221206", "601302", "1634508", "4443055", "12077476", "32829984", "89241150", "242582597", "659407867", "1792456423", "4872401723", "13244561064", "36002449668", "97864804714", "266024120300", "723128532145", "1965667148572" ]

a(n) = floor(cosh(n)).

A000502

[ "42", "1586", "31388", "442610", "5030004", "49145460", "429166584", "3435601554", "25658464260", "181055975100", "1218601601672", "7880146275092", "49238911113224", "298652277299880", "1764885293279472", "10192638073849554", "57674223198273444", "320430129184331628", "1751190732477786600", "9428906326013866076" ]

Number of genus 0 rooted maps with 6 faces and n vertices.

A000503

[ "0", "1", "-3", "-1", "1", "-4", "-1", "0", "-7", "-1", "0", "-226", "-1", "0", "7", "-1", "0", "3", "-2", "0", "2", "-2", "0", "1", "-3", "-1", "1", "-4", "-1", "0", "-7", "-1", "0", "-76", "-1", "0", "7", "-1", "0", "3", "-2", "0", "2", "-2", "0", "1", "-3", "-1", "1", "-4", "-1", "0", "-7", "-1", "0", "-46", "-1", "0", "8", "-1", "0", "3", "-2", "0", "2", "-2", "0", "1", "-3", "-1", "1", "-4", "-1", "0", "-6", "-1", "0", "-33", "-1", "0", "9", "-1", "0", "3", "-2", "0", "2", "-2", "0", "1", "-2", "-1", "1", "-3", "-1", "0", "-6", "-1", "0", "-26" ]

a(n) = floor(tan(n)).

A000504

[ "1", "56", "1918", "56980", "1636635", "47507460", "1422280860", "44346982680", "1446733012725", "49473074851200", "1774073543492250", "66681131440423500", "2624634287988087375", "108060337458000427500", "4647703259223579555000", "208548093035794902390000", "9749651260035434678555625" ]

S2(j,2j+3) where S2(n,k) is a 2-associated Stirling number of the second kind.

A000505

[ "1", "57", "1191", "15619", "156190", "1310354", "9738114", "66318474", "423281535", "2571742175", "15041229521", "85383238549", "473353301060", "2575022097600", "13796160184500", "73008517581444", "382493246941965", "1987497491971605", "10258045633638475" ]

Eulerian numbers (Euler's triangle: column k=5 of A008292, column k=4 of A173018).

A000506

[ "61", "841", "7311", "51663", "325446", "1910706", "10715506", "58258210", "309958755", "1623847695", "8412276585", "43220104041", "220683627988", "1121561317408", "5679711010548", "28683869195556", "144552802373145", "727271783033445" ]

One half of the number of permutations of [n] such that the differences have 5 runs with the same signs.

A000507

[ "61", "1385", "19028", "206276", "1949762", "16889786", "137963364", "1081702420", "8236142455", "61386982075", "450403628440", "3266265481144", "23480284103492", "167687984079924", "1191656966048088", "8436830209386360", "59563995267159825", "419628657826253805" ]

Number of permutations of [n] with exactly 3 increasing runs of length at least 2.

A000508

[ "61", "2763", "38528", "249856", "1066590", "3487246", "9493504", "22634496", "48649086", "96448478", "179369856", "315621376", "530788622", "860061996", "1346126848", "2046820352", "3038120316", "4403100222", "6254596992", "8737505280", "11992903772" ]

Generalized class numbers.

A000509

[ "6", "6", "8", "10", "12", "13", "14", "14", "17", "21", "22", "24" ]

Size of second largest n-arc in PG(2,q), where q runs through the primes and prime powers >= 7.

A000510

[ "7", "9", "9", "11", "15", "15", "17" ]

Maximal number of points in PG(2,q) with at most 3 on a line (next term is 21 or 22).

A000511

[ "1", "1", "2", "3", "5", "8", "11", "17", "25", "33", "47", "67", "87", "117", "160", "207", "270", "356", "455", "584", "751", "945", "1195", "1513", "1882", "2345", "2927", "3608", "4446", "5483", "6701", "8180", "9986", "12109", "14664", "17750", "21371", "25694", "30872", "36937", "44127", "52672", "62658", "74429", "88327", "104524", "123518", "145819", "171737", "201990", "237332", "278289", "325901", "381278", "445272", "519381", "605230", "704170", "818357", "950150", "1101634", "1275907", "1476384", "1706226", "1969869", "2272224", "2618007", "3013559", "3465917", "3982025", "4570898", "5242569", "6007170", "6877474", "7867709", "8992510", "10269905", "11719991", "13363733", "15226469", "17336450", "19723485", "22423058", "25474712", "28920541", "32810028", "37198284", "42144403", "47717124", "53992936", "61054313", "68996364", "77924848", "87954283", "99215750", "111854888" ]

Number of n-step spiral self-avoiding walks on hexagonal lattice, where at each step one may continue in same direction or make turn of 2*Pi/3 counterclockwise.

A000512

[ "0", "0", "1", "1", "2", "7", "16", "51", "224", "1165", "7454", "56349", "481309", "4548786", "46829325", "519812910", "6177695783", "78190425826", "1049510787100", "14886252250208", "222442888670708", "3492326723315796", "57468395960854710", "989052970923320185", "17767732298980160822", "332572885090541084172", "6475438355244504235759", "130954580036269713385884" ]

Number of equivalence classes of n X n matrices over {0,1} with rows and columns summing to 3, where equivalence is defined by row and column permutations.

A000513

[ "0", "0", "0", "1", "1", "4", "16", "194", "3529", "121790", "5582612", "317579783", "21543414506", "1711281449485", "157117486414656", "16502328443493967", "1965612709107379155", "263512349078757245789", "39497131936385398782814", "6579940884199010139737829", "1211896874083479131415289345", "245593008009270037388205883048" ]

Number of equivalence classes of n X n matrices over {0,1} with rows and columns summing to 4, where equivalence is defined by row and column permutations. Also number of isomorphism classes of bicolored quartic bipartite graphs, where isomorphism cannot exchange the colors.

A000514

[ "1", "120", "4293", "88234", "1310354", "15724248", "162512286", "1505621508", "12843262863", "102776998928", "782115518299", "5717291972382", "40457344748072", "278794377854832", "1879708669896492", "12446388300682056", "81180715002105741" ]

Eulerian numbers (Euler's triangle: column k=6 of A008292, column k=5 of A173018)

A000515

[ "1", "12", "180", "2800", "44100", "698544", "11099088", "176679360", "2815827300", "44914183600", "716830370256", "11445589052352", "182811491808400", "2920656969720000", "46670906271240000", "745904795339462400", "11922821963004219300", "190600129650794094000", "3047248986392325330000" ]

a(n) = (2n)!(2n+1)!/n!^4, or equally (2n+1)*binomial(2n,n)^2.

A000516

[ "0", "0", "0", "0", "1", "1", "4", "51", "3529", "601055", "156473848", "54062069505", "23869437984682", "13186966476208771", "8971034249976338907", "7414924597575224629299", "7360058177440420943520750", "8683626883245180573511018830", "12066478410398147578519948851818", "19585444567548740264243478805318202" ]

Number of equivalence classes of n X n matrices over {0,1} with rows and columns summing to 5, where equivalence is defined by row and column permutations. Isomorphism classes of bicolored 5-regular bipartite graphs, where isomorphism cannot exchange the colors.

A000517

[ "272", "7936", "137216", "1841152", "21253376", "222398464", "2174832640", "20261765120", "182172651520", "1594922762240", "13684856848384", "115620218667008", "965271355195392", "7984436548730880", "65569731961159680", "535438370914959360", "4353038473793372160", "35266789418949672960" ]

Number of permutations of length n with exactly three valleys.

A000518

[ "272", "24611", "515086", "4456448", "23750912", "93241002", "296327464", "806453248", "1951153920", "4300685074", "8787223186", "16878338048", "30768878848", "53624926972", "89982082488", "146028888064", "230022888960", "353194774434", "529896144586" ]

Generalized tangent numbers d_(n,4).

A000519

[ "1", "2", "3", "5", "7", "18", "43", "313", "7525", "846992", "324127859", "403254094631", "1555631972009429", "19731915624463099552", "791773335030637885025287", "107432353216118868234728540267", "47049030539260648478475949282317451", "71364337698829887974206671525372672234854" ]

Number of equivalence classes of nonzero regular 0-1 matrices of order n.

A000520

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

Nearest integer to log_10(n).

A000521

[ "1", "744", "196884", "21493760", "864299970", "20245856256", "333202640600", "4252023300096", "44656994071935", "401490886656000", "3176440229784420", "22567393309593600", "146211911499519294", "874313719685775360", "4872010111798142520", "25497827389410525184", "126142916465781843075" ]

Coefficients of modular function j as power series in q = e^(2 Pi i t). Another name is the elliptic modular invariant J(tau).

A000522

[ "1", "2", "5", "16", "65", "326", "1957", "13700", "109601", "986410", "9864101", "108505112", "1302061345", "16926797486", "236975164805", "3554627472076", "56874039553217", "966858672404690", "17403456103284421", "330665665962404000", "6613313319248080001", "138879579704209680022", "3055350753492612960485", "70273067330330098091156" ]

Total number of ordered k-tuples (k=0..n) of distinct elements from an n-element set: a(n) = Sum_{k=0..n} n!/k!.

A000523

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

a(n) = floor(log_2(n)).

A000524

[ "2", "9", "34", "119", "401", "1316", "4247", "13532", "42712", "133816", "416770", "1291731", "3987444", "12266845", "37627230", "115125955", "351467506", "1070908135", "3257389088", "9892759091", "30002923380", "90879555521", "274963755791", "831064788976" ]

Number of rooted trees with n nodes, 2 of which are labeled.

A000525

[ "64", "625", "4016", "21256", "100407", "439646", "1823298", "7258228", "27983518", "105146732", "386812476", "1398023732", "4977320988", "17492710572", "60790051789", "209179971147", "713533304668", "2415061934763", "8117293752058", "27111950991825", "90039381031273" ]

Number of partially labeled rooted trees with n nodes (4 of which are labeled).

A000526

[ "125", "1296", "8716", "47787", "232154", "1040014", "4395772", "17781210", "69498964", "264248924", "982218072", "3582421612", "12857819052", "45515994861", "159205157535", "551049504784", "1889714853263", "6427147635062", "21698583468717" ]

Number of partially labeled trees with n nodes (5 of which are labeled).

A000527

[ "52", "472", "3224", "18888", "101340", "511120", "2465904", "11496144", "52165892", "231557064", "1009247192", "4331502840", "18346242492", "76822836544", "318485778848", "1308750158016", "5335993098340", "21603437175288", "86912657626392", "347660876627944", "1383457374046444", "5479086968052912", "21604984733546336", "84850331177724944", "332001521469767940", "1294589169323791912", "5031934808360234760", "19500424806065865400", "75360646947991208396", "290478417300879735680", "1116919455364101145920", "4284817000807140094464", "16402243457215852326116", "62659647762404302956856", "238910441445219175239480" ]

Series-parallel numbers.

A000528

[ "1", "1", "1", "2", "2", "17", "324", "842227", "57810418543", "104452188344901572", "6108088657705958932053657" ]

Number of types of Latin squares of order n. Equivalently, number of nonisomorphic 1-factorizations of K_{n,n}.

A000529

[ "20", "74", "186", "388", "721", "1236", "1995", "3072", "4554", "6542", "9152", "12516", "16783", "22120", "28713", "36768", "46512", "58194", "72086", "88484", "107709", "130108", "156055", "185952", "220230", "259350", "303804", "354116", "410843", "474576", "545941", "625600", "714252", "812634", "921522", "1041732", "1174121", "1319588", "1479075", "1653568", "1844098", "2051742", "2277624", "2522916", "2788839", "3076664", "3387713", "3723360", "4085032", "4474210" ]

Powers of rooted tree enumerator.

A000530

[ "1", "5", "28", "226", "2077", "20770", "222884", "2529541" ]

Let p(n, s, x) be predicate that number of occurrences of s's in x >= 2*n - the length of the longest sequence of s's in x. Then a(n)=#{x in {0,1}* | x ends in 0 and p(n,0,x) and (there is no prefix y of x such that p(n,0,y) or p(n,1,y))}.

A000531

[ "1", "7", "38", "187", "874", "3958", "17548", "76627", "330818", "1415650", "6015316", "25413342", "106853668", "447472972", "1867450648", "7770342787", "32248174258", "133530264682", "551793690628", "2276098026922", "9373521044908", "38546133661492" ]

From area of cyclic polygon of 2n + 1 sides.

A000532

[ "1", "1", "2", "8", "86", "1770", "88418", "8934966", "2087813834", "1013346943033", "1111598871478668", "2568944901392936854", "13251059359839620127088", "145194816279817259193401518", "3524171261632305641165676374930" ]

Number of Hamiltonian paths from NW to SW corners in an n X n grid.

A000533

[ "1", "11", "101", "1001", "10001", "100001", "1000001", "10000001", "100000001", "1000000001", "10000000001", "100000000001", "1000000000001", "10000000000001", "100000000000001", "1000000000000001", "10000000000000001" ]

a(0)=1; a(n) = 10^n + 1, n >= 1.

A000534

[ "0", "1", "2", "3", "5", "6", "8", "9", "11", "14", "17", "24", "29", "32", "41", "56", "96", "128", "224", "384", "512", "896", "1536", "2048", "3584", "6144", "8192", "14336", "24576", "32768", "57344", "98304", "131072", "229376", "393216", "524288", "917504", "1572864", "2097152", "3670016", "6291456", "8388608", "14680064" ]

Numbers that are not the sum of 4 nonzero squares.

A000535

[ "0", "27", "378", "4536", "48600", "489780", "4738104", "44535456", "409752432", "3708359550", "33125746500", "292779558720", "2565087894720", "22307854940280", "192788833482000", "1657111548654720", "14176605442521312", "120779466450505758", "1025230099571720676", "8674221270307971600" ]

Card matching: coefficients B[n,2] of t^2 in the reduced hit polynomial A[n,n,n](t).

A000536

[ "24", "240", "2520", "26880", "304080", "3671136", "47391120", "653463360", "9603708840", "150046937040", "2485510331304", "43536519673920", "804343214307360", "15636586027419840", "319143375070100640", "6824486562845878656", "152599994618389811640", "3561710724832153990320", "86627571138529803385080", "2192153071078356814538880", "57633178354598014299807984", "1572073330365520093029415200", "44434609885866805678475703600", "1299879247128621094998213278400", "39312834919322919649653205283400", "1227895179113516869799082638629776", "39569125440836907870479047149487560", "1314368274045259508166257769617810880", "44963797526832537006635800892057862720", "1582832153412276057834241761650127323520" ]

Number of 3-line Latin rectangles.

A000537

[ "0", "1", "9", "36", "100", "225", "441", "784", "1296", "2025", "3025", "4356", "6084", "8281", "11025", "14400", "18496", "23409", "29241", "36100", "44100", "53361", "64009", "76176", "90000", "105625", "123201", "142884", "164836", "189225", "216225", "246016", "278784", "314721", "354025", "396900", "443556", "494209", "549081" ]

Sum of first n cubes; or n-th triangular number squared.

A000538

[ "0", "1", "17", "98", "354", "979", "2275", "4676", "8772", "15333", "25333", "39974", "60710", "89271", "127687", "178312", "243848", "327369", "432345", "562666", "722666", "917147", "1151403", "1431244", "1763020", "2153645", "2610621", "3142062", "3756718", "4463999", "5273999", "6197520", "7246096", "8432017", "9768353" ]

Sum of fourth powers: 0^4 + 1^4 + ... + n^4.

A000539

[ "0", "1", "33", "276", "1300", "4425", "12201", "29008", "61776", "120825", "220825", "381876", "630708", "1002001", "1539825", "2299200", "3347776", "4767633", "6657201", "9133300", "12333300", "16417401", "21571033", "28007376", "35970000", "45735625", "57617001", "71965908", "89176276", "109687425", "133987425" ]

Sum of 5th powers: 0^5 + 1^5 + 2^5 + ... + n^5.

A000540

[ "0", "1", "65", "794", "4890", "20515", "67171", "184820", "446964", "978405", "1978405", "3749966", "6735950", "11562759", "19092295", "30482920", "47260136", "71397705", "105409929", "152455810", "216455810", "302221931", "415601835", "563637724", "754740700", "998881325", "1307797101", "1695217590" ]

Sum of 6th powers: 0^6 + 1^6 + 2^6 + ... + n^6.

A000541

[ "0", "1", "129", "2316", "18700", "96825", "376761", "1200304", "3297456", "8080425", "18080425", "37567596", "73399404", "136147921", "241561425", "412420800", "680856256", "1091194929", "1703414961", "2597286700", "3877286700", "5678375241", "8172733129", "11577558576", "16164030000", "22267545625" ]

Sum of 7th powers: 1^7 + 2^7 + ... + n^7.

A000542

[ "0", "1", "257", "6818", "72354", "462979", "2142595", "7907396", "24684612", "67731333", "167731333", "382090214", "812071910", "1627802631", "3103591687", "5666482312", "9961449608", "16937207049", "27957167625", "44940730666", "70540730666", "108363590027", "163239463563", "241550448844" ]

Sum of 8th powers: 1^8 + 2^8 + ... + n^8.

A000543

[ "0", "1", "23", "333", "2916", "16725", "70911", "241913", "701968", "1798281", "4173775", "8942021", "17930628", "34009053", "61518471", "106823025", "179003456", "290715793", "459239463", "707740861", "1066780100", "1576090341", "2286660783", "3263156073", "4586706576" ]

Number of inequivalent ways to color vertices of a cube using at most n colors.

A000544

[ "3", "25", "155", "1005", "7488", "64164", "619986", "6646750", "78161249", "999473835", "13801761213", "204631472475", "3241541125110", "54629642149630", "975867376041308", "18416844056075364", "366128842105397631", "7647337600268371485", "167424323805645018159", "3833790834030516355705", "91641405910147125954428", "2282611988081527293910920" ]

Number of permutations of length n by rises.

A000545

[ "1", "96", "9099", "280832", "4073375", "36292320", "230719293", "1145393152", "4707296613", "16666924000", "52307593239", "148602435840", "388302646355", "944900450144", "2162441849625", "4691253854208", "9710376716137", "19280531603808", "36888593841475", "68266682784000", "122597146773927" ]

Number of ways of n-coloring a dodecahedron.

A000546

[ "1", "12", "28", "314", "98", "386", "943", "1494", "1680", "4722", "6576", "11696", "3982", "2987", "17548", "36208", "7083", "59692", "159116", "79592", "57857", "212160", "352258", "221185", "57346", "294913", "252548", "530052", "331778", "524289", "1088129", "913319", "2065786", "1541308", "1032875", "1264924", "8151894" ]

First occurrence of n consecutive numbers that take same number of steps to reach 1 in 3x+1 problem.

A000547

[ "0", "9", "18", "37", "25", "120", "36", "47", "42", "59", "137", "143", "51", "48", "141", "41", "57", "73", "77", "76", "166", "80", "104", "93", "78", "96", "181", "102", "91", "102", "209", "201", "197", "194", "196", "129", "230", "115", "200", "97", "213", "206", "115", "216", "220", "214", "208", "246", "211", "233", "199", "208", "209", "224", "207", "228" ]

Number of steps to reach 1 in sequence A000546.

A000548

[ "1", "4", "9", "16", "36", "49", "64", "81", "121", "144", "196", "256", "324", "361", "441", "484", "529", "576", "729", "784", "961", "1024", "1089", "1296", "1444", "1764", "1849", "1936", "2116", "2209", "2304", "2401", "2916", "3136", "3249", "3481", "3844", "3969", "4096", "4356", "4489" ]

Squares that are not the sum of 2 nonzero squares.

A000549

[ "1", "2", "4", "5", "8", "10", "13", "16", "20", "25", "32", "37", "40", "52", "58", "64", "80", "85", "100", "128", "130", "148", "160", "208", "232", "256", "320", "340", "400", "512", "520", "592", "640", "832", "928", "1024", "1280", "1360", "1600", "2048", "2080", "2368", "2560", "3328", "3712", "4096", "5120", "5440", "6400", "8192", "8320", "9472", "10240" ]

Numbers that are the sum of 2 squares but not sum of 3 nonzero squares.

A000550

[ "1", "3", "14", "42", "128", "334", "850", "2010", "4625", "10201", "21990", "46108", "94912", "191562", "380933", "746338", "1444676", "2763931", "5235309", "9822686", "18275648", "33734658", "61826344", "112550305", "203627610", "366267931", "655261559", "1166312530", "2066048261", "3643352362", "6397485909", "11188129665", "19491131627", "33831897511", "58519577756", "100885389220", "173368983090", "297021470421", "507378371670", "864277569606", "1468245046383", "2487774321958", "4204663810414", "7089200255686", "11924621337321", "20012746962064", "33513139512868", "56001473574091", "93387290773141", "155419866337746" ]

Number of trees of diameter 7.

A000551

[ "6", "36", "200", "1170", "7392", "50568", "372528", "2936070", "24617120", "218521116", "2045278248", "20112821274", "207162957120", "2228888801040", "24989309310944", "291322555295886", "3524580202643136", "44176839081266340", "572725044269255640" ]

Number of labeled rooted trees of height 2 with n nodes.

A000552

[ "24", "300", "3360", "38850", "475776", "6231960", "87530400", "1316954430", "21173760960", "362670636900", "6596214691248", "126980000240730", "2579214238608000", "55118036257959600", "1235935135837111104", "29009023670878484598" ]

Number of labeled rooted trees of height 3 with n nodes.

A000553

[ "120", "2520", "43680", "757680", "13747104", "264181680", "5395040640", "117080049240", "2696387899920", "65774992411128", "1695845836077120", "46110625382246880", "1319345179723609920", "39640903618873667040", "1248193457738661143808" ]

Number of labeled rooted trees of height 4 with n nodes.

A000554

[ "12", "60", "210", "630", "1736", "4536", "11430", "28050", "67452", "159588", "372554", "859950", "1965840", "4456176", "10026702", "22412970", "49806980", "110100060", "242220594", "530578950", "1157627352", "2516581800", "5452594550", "11777604930", "25367149836", "54492396756", "116769422490", "249644973150" ]

Number of labeled trees of diameter 3 with n nodes.

A000555

[ "60", "720", "6090", "47040", "363384", "2913120", "24560910", "218386080", "2044958916", "20112075984", "207161237010", "2228884869120", "24989300398320", "291322535242176", "3524580157816854", "44176838981652000", "572725044049055100", "7668896804089696560", "105920137922879314650", "1507138839384235136640", "22068265782102952223400", "332178010291171425732000", "5135009134117954527323550", "81449458937043220255508640" ]

Number of labeled trees of diameter 4 with n nodes.

A000556

[ "1", "1", "5", "31", "257", "2671", "33305", "484471", "8054177", "150635551", "3130337705", "71556251911", "1784401334897", "48205833997231", "1402462784186105", "43716593539939351", "1453550100421124417", "51350258701767067711", "1920785418183176050505", "75839622064482770570791" ]

Expansion of exp(-x) / (1 - exp(x) + exp(-x)).

A000557

[ "1", "2", "8", "50", "416", "4322", "53888", "783890", "13031936", "243733442", "5064992768", "115780447730", "2887222009856", "77998677862562", "2269232452763648", "70734934220015570", "2351893466832306176", "83086463910558199682", "3107896091715557654528", "122711086194279627711410" ]

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

A000558

[ "1", "6", "32", "175", "1012", "6230", "40819", "283944", "2090424", "16235417", "132609666", "1135846062", "10175352709", "95108406130", "925496853980", "9357279554071", "98118527430960", "1065259283215810", "11956366813630835", "138539436100687988", "1655071323662574756", "20361556640795422729" ]

Generalized Stirling numbers of second kind.

A000559

[ "1", "12", "110", "945", "8092", "70756", "638423", "5971350", "57996774", "585092607", "6128147610", "66579524648", "749542556193", "8733648533696", "105203108066962", "1308549777461505", "16787682400875456", "221901108871482760", "3018891886411332135", "42230736603244134242" ]

Generalized Stirling numbers of second kind.

A000560

[ "1", "2", "5", "12", "33", "87", "252", "703", "2105", "6099", "18689", "55639", "173423", "526937", "1664094", "5137233", "16393315", "51255709", "164951529", "521138861", "1688959630", "5382512216", "17547919924", "56335234064", "184596351277", "596362337295", "1962723402375" ]

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

A000561

[ "6", "44", "145", "336", "644", "1096", "1719", "2540", "3586", "4884", "6461", "8344", "10560", "13136", "16099", "19476", "23294", "27580", "32361", "37664", "43516", "49944", "56975", "64636", "72954", "81956", "91669", "102120", "113336", "125344", "138171", "151844", "166390", "181836", "198209", "215536", "233844", "253160", "273511", "294924", "317426", "341044" ]