christopher/oeis · Datasets at Hugging Face

a-number stringlengths 7 7	sequence sequencelengths 1 377	description stringlengths 3 852
A000401	[ "0", "1", "2", "3", "4", "5", "6", "7", "8", "9", "10", "11", "12", "13", "15", "16", "17", "18", "19", "20", "21", "22", "23", "24", "25", "26", "27", "28", "29", "31", "32", "33", "34", "35", "36", "37", "38", "39", "40", "41", "42", "43", "44", "45", "47", "48", "49", "50", "51", "52", "53", "54", "55", "57", "58", "59", "60", "61", "63", "64", "65", "66", "67", "68", "69", "70", "71", "72", "73", "74", "75", "76" ]	Numbers of form x^2 + y^2 + 2z^2.
A000402	[ "0", "0", "1", "6", "41", "293", "2309", "19975", "189524", "1960041", "21993884", "266361634", "3465832370", "48245601976", "715756932697", "11277786883720", "188135296651083", "3313338641692957", "61444453534759589", "1196988740015236617", "24442368179977776766", "522124104504306695929" ]	Number of permutations of [n] in which the longest increasing run has length 3.
A000403	[ "1", "1", "2", "2", "5", "6" ]	Number of simple equifacetted 3-manifolds with n faces.
A000404	[ "2", "5", "8", "10", "13", "17", "18", "20", "25", "26", "29", "32", "34", "37", "40", "41", "45", "50", "52", "53", "58", "61", "65", "68", "72", "73", "74", "80", "82", "85", "89", "90", "97", "98", "100", "101", "104", "106", "109", "113", "116", "117", "122", "125", "128", "130", "136", "137", "145", "146", "148", "149", "153", "157", "160", "162", "164", "169", "170", "173", "178" ]	Numbers that are the sum of 2 nonzero squares.
A000405	[ "1", "1", "6", "51", "561", "7556", "120196", "2201856", "45592666", "1051951026", "26740775306", "742069051906", "22310563733864", "722108667742546", "25024187820786357", "924161461265888370", "36223781285638309482", "1501552062016443881514" ]	Number of 6-level labeled rooted trees with n leaves.
A000406	[ "1", "6", "57", "741", "12244", "245755", "5809875", "158198200", "4877852505", "168055077875", "6400217406500", "267058149580823", "12118701719205803", "594291742526530761", "31323687504696772151", "1766116437541895988303", "106080070002238888908150" ]	Coefficients of iterated exponentials.
A000407	[ "1", "6", "60", "840", "15120", "332640", "8648640", "259459200", "8821612800", "335221286400", "14079294028800", "647647525324800", "32382376266240000", "1748648318376960000", "101421602465863680000", "6288139352883548160000", "415017197290314178560000" ]	a(n) = (2*n+1)! / n!.
A000408	[ "3", "6", "9", "11", "12", "14", "17", "18", "19", "21", "22", "24", "26", "27", "29", "30", "33", "34", "35", "36", "38", "41", "42", "43", "44", "45", "46", "48", "49", "50", "51", "53", "54", "56", "57", "59", "61", "62", "65", "66", "67", "68", "69", "70", "72", "73", "74", "75", "76", "77", "78", "81", "82", "83", "84", "86", "88", "89", "90", "91", "93", "94", "96", "97", "98", "99", "101", "102", "104" ]	Numbers that are the sum of three nonzero squares.
A000409	[ "0", "6", "350", "43260", "14591171", "14657461469", "46173502811223", "474928141312623525", "16489412944755088235117", "1985178211854071817861662307", "846428472480689964807653763864449", "1299141117072945982773752362381072143359", "7268140170419155675761326840423792818571154945", "149650282980396792665043455999899697765782372693740287" ]	Singular n X n (0,1)-matrices: the number of n X n (0,1)-matrices having distinct, nonzero ordered rows, but having at least two equal columns or at least one zero column.
A000410	[ "0", "0", "6", "425", "65625", "27894671", "35716401889", "144866174953833" ]	Number of singular n X n rational (0,1)-matrices.
A000411	[ "6", "522", "152166", "93241002", "97949265606", "157201459863882", "357802951084619046", "1096291279711115037162", "4350684698032741048452486", "21709332137467778453687752842", "133032729004732721625426681085926", "982136301747914281420205946546842922", "8597768767880274820173388403096814519366" ]	Generalized tangent numbers d(6,n).
A000412	[ "3", "7", "16", "31", "57", "97", "162", "257", "401", "608", "907", "1325", "1914", "2719", "3824", "5313", "7316", "9973", "13495", "18105", "24132", "31938", "42021", "54948", "71484", "92492", "119120", "152686", "194887", "247693", "313613", "395547", "497154", "622688", "777424", "967525", "1200572", "1485393", "1832779", "2255317" ]	Number of bipartite partitions of n white objects and 3 black ones.
A000413	[ "1", "7", "19", "57", "81", "251", "437", "691", "739", "1743", "3695", "6619", "8217", "9771", "14771", "15155", "16831", "18805", "26745", "30551", "41755", "46297", "54339", "72359", "86407", "96969", "131059", "344859", "395231", "519963", "607141", "677397", "741509", "893019", "917217", "1288415", "1406811", "1789599", "1827927", "3085785", "3216051", "3444439", "3524869" ]	Let A(n) = #{(i,j,k): i^2 + j^2 + k^2 <= n}, V(n) = (4/3)Pin^(3/2), P(n) = A(n) - V(n); A000092 gives values of n where \|P(n)\| sets a new record; sequence gives A(A000092(n)).
A000414	[ "4", "7", "10", "12", "13", "15", "16", "18", "19", "20", "21", "22", "23", "25", "26", "27", "28", "30", "31", "33", "34", "35", "36", "37", "38", "39", "40", "42", "43", "44", "45", "46", "47", "48", "49", "50", "51", "52", "53", "54", "55", "57", "58", "59", "60", "61", "62", "63", "64", "65", "66", "67", "68", "69", "70", "71", "72", "73", "74", "75", "76", "77", "78", "79", "80", "81" ]	Numbers that are the sum of 4 nonzero squares.
A000415	[ "2", "5", "8", "10", "13", "17", "18", "20", "26", "29", "32", "34", "37", "40", "41", "45", "50", "52", "53", "58", "61", "65", "68", "72", "73", "74", "80", "82", "85", "89", "90", "97", "98", "101", "104", "106", "109", "113", "116", "117", "122", "125", "128", "130", "136", "137", "145", "146", "148", "149", "153", "157", "160", "162", "164", "170", "173", "178", "180", "181" ]	Numbers that are the sum of 2 but no fewer nonzero squares.
A000416	[ "1", "7", "28", "105", "357", "1197", "3857", "12300", "38430", "118874", "362670", "1095430", "3271751", "9673993", "28310881", "82033609", "235359901", "668779076", "1882412994", "5249817573", "14510628853" ]	Number of 6-dimensional partitions of n.
A000417	[ "1", "7", "28", "105", "357", "1232", "4067", "13301", "42357", "132845", "409262", "1243767", "3727360", "11036649", "32300795", "93538278", "268164868", "761656685", "2144259516", "5986658951", "16583102077", "45593269265", "124464561544", "337479729179", "909156910290", "2434121462871", "6478440788169" ]	Euler transform of A000389.
A000418	[ "0", "0", "0", "0", "0", "0", "0", "1", "7", "34", "136", "487", "1615", "5079", "15349", "45009", "128899", "362266", "1002681", "2740448", "7411408", "19865445", "52840977", "139624510", "366803313", "958696860", "2494322662", "6463281890", "16686206047", "42935345688", "110142163940", "281763465941" ]	Number of n-node rooted trees of height 7.
A000419	[ "3", "6", "11", "12", "14", "19", "21", "22", "24", "27", "30", "33", "35", "38", "42", "43", "44", "46", "48", "51", "54", "56", "57", "59", "62", "66", "67", "69", "70", "75", "76", "77", "78", "83", "84", "86", "88", "91", "93", "94", "96", "99", "102", "105", "107", "108", "110", "114", "115", "118", "120", "123", "126", "129", "131", "132", "133", "134", "138", "139", "140", "141", "142" ]	Numbers that are the sum of 3 but no fewer nonzero squares.
A000420	[ "1", "7", "49", "343", "2401", "16807", "117649", "823543", "5764801", "40353607", "282475249", "1977326743", "13841287201", "96889010407", "678223072849", "4747561509943", "33232930569601", "232630513987207", "1628413597910449", "11398895185373143", "79792266297612001", "558545864083284007" ]	Powers of 7: a(n) = 7^n.
A000421	[ "1", "2", "6", "20", "91", "509", "3608", "31856", "340416", "4269971", "61133757", "978098997", "17228295555", "330552900516", "6853905618223", "152626436936272", "3631575281503404", "91928898608055819", "2466448432564961852", "69907637101781318907" ]	Number of isomorphism classes of connected 3-regular (trivalent, cubic) loopless multigraphs of order 2n.
A000422	[ "1", "21", "321", "4321", "54321", "654321", "7654321", "87654321", "987654321", "10987654321", "1110987654321", "121110987654321", "13121110987654321", "1413121110987654321", "151413121110987654321", "16151413121110987654321", "1716151413121110987654321", "181716151413121110987654321" ]	Concatenation of numbers from n down to 1.
A000423	[ "2", "3", "6", "12", "18", "24", "36", "48", "54", "72", "96", "108", "144", "162", "192", "216", "288", "324", "384", "432", "486", "576", "648", "768", "864", "972", "1152", "1296", "1458", "1536", "1728", "1944", "2304", "2592", "2916", "3072", "3456", "3888", "4374", "4608", "5184", "5832", "6144", "6912", "7776", "8748", "9216", "10368", "11664" ]	a(n) is smallest number > a(n-1) of form a(i)*a(j), i < j < n.
A000424	[ "7", "85", "1660", "48076", "1942416", "104587344", "7245893376", "628308907776", "66687811660800", "8506654697548800", "1284292319599411200", "226530955276874956800", "46165213716463676620800", "10765453901922078105600000", "2848453606917036402278400000", "848800150518516674081587200000" ]	Differences of reciprocals of unity.
A000425	[ "2", "0", "0", "8", "30", "192", "1344", "10800", "97434", "976000", "10749024", "129103992", "1679495350", "23525384064", "353028802560", "5650370001120", "96082828074162", "1729886440780800", "32874134679574208", "657589108734075240", "13811277748363437006", "303884178002526338624" ]	Coefficients of ménage hit polynomials.
A000426	[ "0", "1", "1", "1", "8", "35", "211", "1459", "11584", "103605", "1030805", "11291237", "135015896", "1749915271", "24435107047", "365696282855", "5839492221440", "99096354764009", "1780930394412009", "33789956266629001", "674939337282352360", "14157377139256183723", "311135096550816014651" ]	Coefficients of ménage hit polynomials.
A000427	[ "1", "8", "36", "148", "554", "2024", "7134", "24796", "84625", "285784", "953430", "3151332", "10314257", "33457972", "107557792", "342732670", "1082509680", "3389190112", "10518508294", "32361863632" ]	Number of 7-dimensional partitions of n.
A000428	[ "1", "8", "36", "148", "554", "2094", "7624", "27428", "96231", "332159", "1126792", "3769418", "12437966", "40544836", "130643734", "416494314", "1314512589", "4110009734", "12737116845", "39144344587", "119350793207", "361173596536", "1085171968872" ]	Euler transform of A000579.
A000429	[ "0", "0", "0", "0", "0", "0", "0", "0", "1", "8", "43", "188", "728", "2593", "8706", "27961", "86802", "262348", "776126", "2256418", "6466614", "18311915", "51334232", "142673720", "393611872", "1078955836", "2941029334", "7977065816", "21541492856", "57942770689", "155304829763", "414934057486" ]	Number of n-node rooted trees of height 8.
A000430	[ "2", "3", "4", "5", "7", "9", "11", "13", "17", "19", "23", "25", "29", "31", "37", "41", "43", "47", "49", "53", "59", "61", "67", "71", "73", "79", "83", "89", "97", "101", "103", "107", "109", "113", "121", "127", "131", "137", "139", "149", "151", "157", "163", "167", "169", "173", "179", "181", "191", "193", "197", "199", "211", "223" ]	Primes and squares of primes.
A000431	[ "0", "0", "0", "2", "16", "88", "416", "1824", "7680", "31616", "128512", "518656", "2084864", "8361984", "33497088", "134094848", "536608768", "2146926592", "8588754944", "34357248000", "137433710592", "549744803840", "2199000186880", "8796044787712", "35184271425536", "140737278640128", "562949517213696" ]	Expansion of 2x^3/((1-2x)^2(1-4x)).
A000432	[ "8", "52", "288", "1424", "6648", "29700", "128800", "545600", "2269672", "9303140", "37672216", "150998016", "599988696", "2366216164", "9270987656", "36116062832", "139978757920", "540069059028", "2075217121688", "7944690769952", "30313624200640", "115312027433188", "437420730644304", "1655047867097280", "6247339311097296", "23530440547115428", "88447214709073696", "331832490378209152", "1242766581420901656", "4646714574562484628", "17347357264162110368", "64668460220964604944", "240747014238189337840", "895102104022837748484", "3323982608759454833032", "12329573838525875316560", "45684294664598118867184", "169098457957523787786644" ]	Series-parallel numbers.
A000433	[ "0", "1", "2", "3", "4", "5", "6", "7", "10", "11", "12", "13", "14", "15", "16", "17", "20", "21", "22", "23", "24", "25", "26", "27", "30", "31", "32", "100", "101", "102", "103", "104", "105", "106", "107", "110", "111", "112", "113", "114", "115", "116", "117", "120", "121", "122", "123", "124", "125", "126", "127", "130", "131", "132", "200", "201", "202", "203" ]	n written in base where place values are positive cubes.
A000434	[ "0", "0", "0", "1", "8", "67", "602", "5811", "60875", "690729", "8457285", "111323149", "1569068565", "23592426102", "377105857043", "6387313185576", "114303481217657", "2155348564847332", "42719058006864690", "887953677898186108", "19316200230609433690", "438920223893512987430" ]	Number of permutations of [n] in which the longest increasing run has length 4.
A000435	[ "0", "1", "8", "78", "944", "13800", "237432", "4708144", "105822432", "2660215680", "73983185000", "2255828154624", "74841555118992", "2684366717713408", "103512489775594200", "4270718991667353600", "187728592242564421568", "8759085548690928992256", "432357188322752488126152", "22510748754252398927872000" ]	Normalized total height of all nodes in all rooted trees with n labeled nodes.
A000436	[ "1", "8", "352", "38528", "7869952", "2583554048", "1243925143552", "825787662368768", "722906928498737152", "806875574817679474688", "1118389087843083461066752", "1884680130335630169428983808", "3794717805092151129643367268352" ]	Generalized Euler numbers c(3,n).
A000437	[ "0", "9", "41", "81", "146", "194", "306", "369", "425", "594", "689", "866", "1109", "1161", "1154", "1361", "1634", "1781", "1889", "2141", "2729", "2609", "3626", "3366", "3566", "3449", "3506", "4241", "4289", "4826", "5066", "5381", "7034", "5561", "6254", "7229", "7829", "8186", "8069", "8126", "8609", "8921", "8774" ]	Smallest nonnegative number that is the sum of 3 squares in exactly n ways.
A000438	[ "1", "1", "6", "6240", "1225566720", "252282619805368320", "98758655816833727741338583040" ]	Number of 1-factorizations of complete graph K_{2n}.
A000439	[ "9", "30", "69", "133", "230", "369", "560", "814", "1143", "1560", "2079", "2715", "3484", "4403", "5490", "6764", "8245", "9954", "11913", "14145", "16674", "19525", "22724", "26298", "30275", "34684", "39555", "44919", "50808", "57255", "64294", "71960", "80289", "89318", "99085", "109629", "120990", "133209", "146328", "160390", "175439", "191520", "208679" ]	Powers of rooted tree enumerator.
A000440	[ "9", "30", "180", "980", "8326", "70272", "695690", "7518720", "89193276", "1148241458", "15947668065", "237613988040", "3780133322620", "63945806121448", "1146081593303784", "21693271558730304", "432411684714253605", "9053476937543082240", "198641103956454088919" ]	Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-4 places.
A000441	[ "0", "1", "9", "34", "95", "210", "406", "740", "1161", "1920", "2695", "4116", "5369", "7868", "9690", "13640", "16116", "22419", "25365", "34160", "38640", "50622", "55154", "73320", "77225", "100100", "107730", "135576", "141085", "182340", "184760", "233616", "243408", "297738", "301420", "385110", "377511", "467210", "478842" ]	a(n) = Sum_{k=1..n-1} ksigma(k)sigma(n-k).
A000442	[ "1", "1", "8", "216", "13824", "1728000", "373248000", "128024064000", "65548320768000", "47784725839872000", "47784725839872000000", "63601470092869632000000", "109903340320478724096000000", "241457638684091756838912000000", "662559760549147780765974528000000", "2236139191853373760085164032000000000" ]	a(n) = (n!)^3.
A000443	[ "325", "425", "625", "650", "725", "845", "850", "925", "1025", "1250", "1300", "1325", "1445", "1450", "1525", "1690", "1700", "1825", "1850", "2050", "2225", "2425", "2500", "2525", "2600", "2650", "2725", "2825", "2873", "2890", "2900", "2925", "3050", "3125", "3380", "3400", "3425", "3650", "3700", "3725", "3757", "3825", "3925", "4100" ]	Numbers that are the sum of 2 squares in exactly 3 ways.
A000444	[ "9", "64", "326", "1433", "5799", "22224", "81987", "293987", "1031298", "3555085", "12081775", "40576240", "134919788", "444805274", "1455645411", "4733022100", "15302145060", "49223709597", "157629612076", "502736717207", "1597541346522", "5059625685739", "15975936032821", "50304490599602" ]	Number of partially labeled rooted trees with n nodes (3 of which are labeled).
A000445	[ "9", "77", "1224", "7888", "202124", "1649375" ]	Latest possible occurrence of the first consecutive pair of n-th power residues, modulo any prime.
A000446	[ "0", "25", "325", "1105", "4225", "5525", "203125", "27625", "71825", "138125", "2640625", "160225", "17850625", "1221025", "1795625", "801125", "1650390625", "2082925", "49591064453125", "4005625", "44890625", "2158203125", "30525625", "5928325", "303460625", "53955078125" ]	Smallest number that is the sum of 2 squares (allowing zeros) in exactly n ways.
A000447	[ "0", "1", "10", "35", "84", "165", "286", "455", "680", "969", "1330", "1771", "2300", "2925", "3654", "4495", "5456", "6545", "7770", "9139", "10660", "12341", "14190", "16215", "18424", "20825", "23426", "26235", "29260", "32509", "35990", "39711", "43680", "47905", "52394", "57155", "62196", "67525", "73150", "79079", "85320", "91881", "98770", "105995", "113564", "121485" ]	a(n) = 1^2 + 3^2 + 5^2 + 7^2 + ... + (2n-1)^2 = n(4*n^2 - 1)/3.
A000448	[ "0", "25", "325", "1105", "4225", "5525", "27625", "27625", "71825", "138125", "160225", "160225", "801125", "801125", "801125", "801125", "2082925", "2082925", "4005625", "4005625", "5928325", "5928325", "5928325", "5928325", "29641625", "29641625", "29641625", "29641625", "29641625", "29641625", "29641625", "29641625" ]	Smallest number that is the sum of 2 squares in at least n ways.
A000449	[ "1", "0", "10", "40", "315", "2464", "22260", "222480", "2447445", "29369120", "381798846", "5345183480", "80177752655", "1282844041920", "21808348713320", "392550276838944", "7458455259940905", "149169105198816960", "3132551209175157490", "68916126601853463240" ]	Rencontres numbers: number of permutations of [n] with exactly 3 fixed points.
A000450	[ "1", "10", "50", "385", "3130", "28764", "291900", "3249210", "39367395", "515874470", "7270929806", "109691447395", "1763782644020", "30114243100760", "544123405603800", "10373304279494964", "208092476689208805", "4381794437494059810", "96635643877147885450", "2227562535558247510885" ]	Coefficients of ménage hit polynomials.
A000451	[ "0", "9", "41", "81", "146", "194", "306", "369", "425", "594", "689", "866", "1109", "1154", "1154", "1361", "1634", "1781", "1889", "2141", "2609", "2609", "3366", "3366", "3449", "3449", "3506", "4241", "4289", "4826", "5066", "5381", "5561", "5561", "6254", "7229", "7829", "8069", "8069", "8126", "8609", "8774", "8774" ]	Smallest number that is the sum of 3 squares in at least n ways.
A000452	[ "1", "2", "3", "5", "6", "7", "8", "10", "11", "13", "14", "15", "16", "17", "19", "21", "22", "23", "24", "26", "27", "29", "30", "31", "33", "34", "35", "37", "38", "39", "40", "41", "42", "43", "46", "47", "48", "51", "53", "54", "55", "56", "57", "58", "59", "61", "62", "65", "66", "67", "69", "70", "71", "73", "74", "77", "78", "79", "80", "81", "82", "83", "85", "86", "87", "88", "89", "91", "93", "94", "95", "97", "101" ]	The greedy sequence of integers which avoids 3-term geometric progressions.
A000453	[ "1", "10", "65", "350", "1701", "7770", "34105", "145750", "611501", "2532530", "10391745", "42355950", "171798901", "694337290", "2798806985", "11259666950", "45232115901", "181509070050", "727778623825", "2916342574750", "11681056634501", "46771289738810", "187226356946265", "749329038535350" ]	Stirling numbers of the second kind, S(n,4).
A000454	[ "1", "10", "85", "735", "6769", "67284", "723680", "8409500", "105258076", "1414014888", "20313753096", "310989260400", "5056995703824", "87077748875904", "1583313975727488", "30321254007719424", "610116075740491776" ]	Unsigned Stirling numbers of first kind s(n,4).
A000455	[ "1", "2", "4", "8", "1", "6", "3", "2", "6", "4", "1", "2", "8", "2", "5", "6", "5", "1", "2", "1", "0", "2", "4", "2", "0", "4", "8", "4", "0", "9", "6", "8", "1", "9", "2", "1", "6", "3", "8", "4", "3", "2", "7", "6", "8", "6", "5", "5", "3", "6", "1", "3", "1", "0", "7", "2", "2", "6", "2", "1", "4", "4", "5", "2", "4", "2", "8", "8", "1", "0", "4", "8", "5", "7", "6", "2", "0", "9", "7", "1", "5", "2" ]	Digits of powers of 2.
A000456	[ "0", "0", "0", "0", "1", "10", "99", "1024", "11304", "133669", "1695429", "23023811", "333840443", "5153118154", "84426592621", "1463941342191", "26793750988542", "516319125748337", "10451197169218523", "221738082618710329", "4921234092461339819", "114041894068935641488" ]	Number of permutations of [n] in which the longest increasing run has length 5.
A000457	[ "1", "10", "105", "1260", "17325", "270270", "4729725", "91891800", "1964187225", "45831035250", "1159525191825", "31623414322500", "924984868933125", "28887988983603750", "959493919812553125", "33774185977401870000", "1255977541034632040625" ]	Exponential generating function: (1+3x)/(1-2x)^(7/2).
A000458	[ "1", "1", "2", "4", "10", "25", "49", "88", "237", "500", "1412", "3570", "12846", "36072", "126504", "493920", "2358720", "12292224", "49984224", "171237888", "642804078", "1980490350", "6380883000", "27032104440", "117961599600", "555861355920", "2623445776224" ]	a(0) = a(1) = 1; thereafter a(n) = sigma(a(n-1)) + sigma(a(n-2)).
A000459	[ "1", "0", "1", "10", "297", "13756", "925705", "85394646", "10351036465", "1596005408152", "305104214112561", "70830194649795010", "19629681235869138841", "6401745422388206166420", "2427004973632598297444857", "1058435896607583305978409166", "526149167104704966948064477665" ]	Number of multiset permutations of {1, 1, 2, 2, ..., n, n} with no fixed points.
A000460	[ "1", "11", "66", "302", "1191", "4293", "14608", "47840", "152637", "478271", "1479726", "4537314", "13824739", "41932745", "126781020", "382439924", "1151775897", "3464764515", "10414216090", "31284590870", "93941852511", "282010106381", "846416194536", "2540053889352", "7621839388981", "22869007827143" ]	Eulerian numbers (Euler's triangle: column k=3 of A008292, column k=2 of A173018).
A000461	[ "1", "22", "333", "4444", "55555", "666666", "7777777", "88888888", "999999999", "10101010101010101010", "1111111111111111111111", "121212121212121212121212", "13131313131313131313131313", "1414141414141414141414141414", "151515151515151515151515151515", "16161616161616161616161616161616" ]	Concatenate n n times.
A000462	[ "1", "2", "10", "11", "12", "100", "101", "102", "110", "1000", "1001", "1002", "1010", "1011", "10000", "10001", "10002", "10010", "10011", "10012", "100000", "100001", "100002", "100010", "100011", "100012", "100100", "1000000", "1000001", "1000002", "1000010", "1000011", "1000012", "1000100", "1000101", "10000000", "10000001" ]	Numbers written in base of triangular numbers.
A000463	[ "1", "1", "2", "4", "3", "9", "4", "16", "5", "25", "6", "36", "7", "49", "8", "64", "9", "81", "10", "100", "11", "121", "12", "144", "13", "169", "14", "196", "15", "225", "16", "256", "17", "289", "18", "324", "19", "361", "20", "400", "21", "441", "22", "484", "23", "529", "24", "576", "25", "625", "26", "676", "27", "729", "28", "784", "29", "841", "30", "900", "31", "961", "32", "1024", "33", "1089", "34", "1156", "35", "1225", "36", "1296" ]	n followed by n^2.
A000464	[ "1", "11", "361", "24611", "2873041", "512343611", "129570724921", "44110959165011", "19450718635716001", "10784052561125704811", "7342627959965776406281", "6023130568334172003579011", "5858598896811701995459355761", "6667317162352419006959182803611", "8776621742176931117228228227924441" ]	Expansion of sin x /cos 2x.
A000465	[ "5", "12", "29", "57", "109", "189", "323", "522", "831", "1279", "1941", "2876", "4215", "6066", "8644", "12151", "16933", "23336", "31921", "43264", "58250", "77825", "103362", "136371", "178975", "233532", "303268", "391831", "504069", "645520", "823419", "1046067", "1324136", "1669950", "2099104", "2629685", "3284325", "4089300" ]	Number of bipartite partitions of n white objects and 4 black ones.
A000466	[ "-1", "3", "15", "35", "63", "99", "143", "195", "255", "323", "399", "483", "575", "675", "783", "899", "1023", "1155", "1295", "1443", "1599", "1763", "1935", "2115", "2303", "2499", "2703", "2915", "3135", "3363", "3599", "3843", "4095", "4355", "4623", "4899", "5183", "5475", "5775", "6083", "6399", "6723", "7055", "7395" ]	a(n) = 4*n^2 - 1.
A000467	[ "0", "0", "0", "0", "0", "1", "12", "137", "1602", "19710", "257400", "3574957", "52785901", "827242933", "13730434111", "240806565782", "4452251786946", "86585391630673", "1767406549387381", "37790452850585180", "844817788372455779", "19711244788916894489", "479203883157602851294" ]	Number of permutations of [n] in which the longest increasing run has length 6.
A000468	[ "1", "12", "144", "1750", "23420", "303240", "3641100", "46113200", "575360400", "7346545000", "112402762000", "1351035564000", "16432451210000", "221411634520000", "2657142036440000", "34327724461500000", "434157115760200000", "5432127413542400000", "67405553164731000000", "1053071060221172000000" ]	Powers of ten written in base 8.
A000469	[ "1", "6", "10", "14", "15", "21", "22", "26", "30", "33", "34", "35", "38", "39", "42", "46", "51", "55", "57", "58", "62", "65", "66", "69", "70", "74", "77", "78", "82", "85", "86", "87", "91", "93", "94", "95", "102", "105", "106", "110", "111", "114", "115", "118", "119", "122", "123", "129", "130", "133", "134", "138", "141", "142", "143", "145", "146", "154", "155", "158" ]	1 together with products of 2 or more distinct primes.
A000470	[ "13", "72", "595", "4096", "39078", "379760", "4181826", "49916448", "647070333", "9035216428", "135236990388", "2159812592384", "36658601139066", "658942295734944", "12504663617290908", "249823152134646144", "5241223014084306270", "115206851288747267148", "2647678812396326064043" ]	Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-5 places.
A000471	[ "0", "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(sinh(n)).
A000472	[ "2", "5", "28", "802", "643726", "414383582242", "171713753231982206218246", "29485613049014079571725771288849499850026859242" ]	a(n) = a(n-1)^2 + (a(n-2) + 1)*(a(n-1) - a(n-2)^2).
A000473	[ "14", "386", "5868", "65954", "614404", "5030004", "37460376", "259477218", "1697186964", "10596579708", "63663115880", "370293754740", "2095108370600", "11574690111400", "62629794691632", "332742342741090", "1739371969822260", "8961709528660140", "45576855706440520", "229087231033907708" ]	Number of genus 0 rooted maps with 5 faces and n vertices.
A000474	[ "1", "1", "1", "6", "396", "526915620", "1132835421602062347" ]	Number of nonisomorphic 1-factorizations of complete graph K_{2n}.
A000475	[ "1", "0", "15", "70", "630", "5544", "55650", "611820", "7342335", "95449640", "1336295961", "20044438050", "320711010620", "5452087178160", "98137569209940", "1864613814984984", "37292276299704525", "783137802293789040", "17229031650463366195", "396267727960657413630" ]	Rencontres numbers: number of permutations of [n] with exactly 4 fixed points.
A000476	[ "15", "72", "609", "4960", "46188", "471660", "5275941", "64146768", "842803767", "11902900380", "179857257960", "2895705788736", "49491631601635", "895010868095256", "17074867330880805", "342733960299356800", "7220616209235766260", "159312370008282356844", "3673720238903201471593" ]	Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-1 places.
A000477	[ "0", "1", "15", "76", "275", "720", "1666", "3440", "6129", "11250", "17545", "28896", "41405", "65072", "85950", "128960", "162996", "238545", "286995", "404600", "482160", "662112", "756470", "1042560", "1150625", "1549730", "1732590", "2257920", "2443105", "3250800", "3421160", "4452096", "4791600", "6039522", "6296500" ]	a(n) = Sum_{k=1..n-1} k^2sigma(k)sigma(n-k).
A000478	[ "15", "105", "490", "1918", "6825", "22935", "74316", "235092", "731731", "2252341", "6879678", "20900922", "63259533", "190957923", "575363776", "1731333808", "5205011031", "15638101281", "46962537810", "140988276150", "423174543025", "1269959836015", "3810785476980", "11434235478348", "34306598748315", "102927849307725" ]	Number of ways of placing n labeled balls into 3 indistinguishable boxes with at least 2 balls in each box.
A000479	[ "1", "1", "1", "2", "24", "1344", "1128960", "12198297600", "2697818265354240", "15224734061278915461120", "2750892211809148994633229926400", "19464657391668924966616671344752852992000" ]	Number of 1-factorizations of K_{n,n}.
A000480	[ "1", "0", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1", "0", "0", "0", "0", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1", "0", "0", "0", "0", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1" ]	a(n) = floor(cos(n)).
A000481	[ "1", "15", "140", "1050", "6951", "42525", "246730", "1379400", "7508501", "40075035", "210766920", "1096190550", "5652751651", "28958095545", "147589284710", "749206090500", "3791262568401", "19137821912055", "96416888184100", "485000783495250", "2436684974110751", "12230196160292565", "61338207158409090" ]	Stirling numbers of the second kind, S(n,5).
A000482	[ "1", "15", "175", "1960", "22449", "269325", "3416930", "45995730", "657206836", "9957703756", "159721605680", "2706813345600", "48366009233424", "909299905844112", "17950712280921504", "371384787345228000", "8037811822645051776", "181664979520697076096", "4280722865357147142912", "105005310755917452984576" ]	Unsigned Stirling numbers of first kind s(n,5).
A000483	[ "15", "210", "2380", "26432", "303660", "3678840", "47324376", "647536032", "9418945536", "145410580224", "2377609752960", "41082721413120", "748459539843840", "14345340443665920", "288650580508961280", "6085390148673177600", "134167064248901376000", "3088040233895705088000", "74077507611407752704000", "1849221425299053367296000" ]	Associated Stirling numbers: second order reciprocal Stirling numbers (Fekete) [[n, 3]]. The number of 3-orbit permutations of an n-set with at least 2 elements in each orbit.
A000484	[ "1", "1", "0", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "0", "1", "1", "1", "0", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "0", "1", "1", "0" ]	a(n) = round(cos(n)).
A000485	[ "16", "125", "680", "3135", "13155", "51873", "195821", "715614", "2550577", "8911942", "30640888", "103951415", "348724844", "1158722880", "3818514232", "12493703403", "40620949971", "131336770375", "422536529249", "1353341880777", "4317248276746", "13722302173753" ]	Number of partially labeled trees with n nodes (4 of which are labeled).
A000486	[ "16", "150", "926", "4788", "22548", "100530", "433162", "1825296", "7577120", "31130190", "126969558", "515183724", "2082553132", "8395437930", "33776903714", "135691891272", "544517772984", "2183315948550", "8748985781230", "35043081823140", "140313684667076" ]	One half of the number of permutations of [n] such that the differences have 4 runs with the same signs.
A000487	[ "16", "272", "2880", "24576", "185856", "1304832", "8728576", "56520704", "357888000", "2230947840", "13754155008", "84134068224", "511780323328", "3100738912256", "18733264797696", "112949304754176", "680032201605120", "4090088616099840", "24582312700149760", "147669797096652800" ]	Number of permutations of length n with exactly two valleys.
A000488	[ "16", "361", "3362", "16384", "55744", "152166", "355688", "739328", "1415232", "2529614", "4261454", "6885376", "10708160", "16054580", "23494584", "33554432", "46698624", "64037790", "86342918", "114163712", "149518720", "193356526", "246232840", "311635968", "390600000" ]	Generalized tangent numbers d_(n,3).
A000489	[ "1", "16", "435", "7136", "99350", "1234032", "14219212", "155251840", "1628202762", "16550991200", "164111079110", "1594594348800", "15235525651840", "143518352447680", "1335670583147400", "12301278983461376", "112264111607438906", "1016361486936571680", "9136254276320346046" ]	Card matching: Coefficients B[n,3] of t^3 in the reduced hit polynomial A[n,n,n](t).
A000490	[ "1", "16", "1280", "249856", "90767360", "52975108096", "45344872202240", "53515555843342336", "83285910482761809920", "165262072909347030040576", "407227428060372417275494400", "1219998300294918683087199010816", "4366953142363907901751614431559680", "18406538229888710811704852978971181056" ]	Generalized Euler numbers c(4,n).
A000491	[ "7", "19", "47", "97", "189", "339", "589", "975", "1576", "2472", "3804", "5727", "8498", "12400", "17874", "25433", "35818", "49908", "68939", "94378", "128234", "172917", "231630", "308240", "407804", "536412", "701910", "913773", "1184022", "1527165", "1961432", "2508762", "3196473", "4057403", "5132066" ]	Number of bipartite partitions of n white objects and 5 black ones.
A000492	[ "20", "154", "1676", "14292", "155690", "1731708", "21264624", "280260864", "3970116255", "60113625680", "969368687752", "16588175089420", "300272980075896", "5733025551810600", "115148956467702600", "2427199940533198992", "53576182138937428377", "1235917889588345408586" ]	Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-6 places.
A000493	[ "0", "0", "0", "0", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1", "0", "0", "0", "0", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1", "0", "0" ]	a(n) = floor(sin(n)).
A000494	[ "0", "1", "1", "0", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "0", "1", "1", "1", "0", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "0", "1", "1", "1", "0", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "0", "1", "1", "0", "-1" ]	Nearest integer to sin(n).
A000495	[ "0", "1", "4", "10", "27", "74", "202", "548", "1490", "4052", "11013", "29937", "81377", "221207", "601302", "1634509", "4443055", "12077476", "32829985", "89241150", "242582598", "659407867", "1792456423", "4872401723", "13244561065", "36002449669", "97864804714", "266024120301", "723128532146", "1965667148572" ]	Nearest integer to sinh(n).
A000496	[ "1", "1", "2", "6", "24", "44", "80", "144", "260", "476", "872", "1600", "2940", "5404", "9936", "18272", "33604", "61804", "113672", "209072", "384540", "707276", "1300880", "2392688", "4400836", "8094396", "14887912", "27383136", "50365436", "92636476", "170385040", "313386944", "576408452", "1060180428", "1949975816", "3586564688" ]	Restricted permutations.
A000497	[ "1", "25", "490", "9450", "190575", "4099095", "94594500", "2343240900", "62199262125", "1764494857125", "53338158823950", "1712934942468750", "58274046742786875", "2094379201311271875", "79318164037837725000", "3157886388887074845000" ]	S2(j,2j+2) where S2(n,k) is a 2-associated Stirling number of the second kind.
A000498	[ "1", "26", "302", "2416", "15619", "88234", "455192", "2203488", "10187685", "45533450", "198410786", "848090912", "3572085255", "14875399450", "61403313100", "251732291184", "1026509354985", "4168403181210", "16871482830550", "68111623139600" ]	Eulerian numbers (Euler's triangle: column k=4 of A008292, column k=3 of A173018)
A000499	[ "0", "1", "27", "184", "875", "2700", "7546", "17600", "35721", "72750", "126445", "223776", "353717", "595448", "843750", "1349120", "1827636", "2808837", "3600975", "5306000", "6667920", "9599172", "11509982", "16416000", "19015625", "26605670", "30902310", "41686848", "46948825", "64233000", "70306760", "94089216" ]	a(n) = Sum_{k=1..n-1} k^3sigma(k)sigma(n-k).
A000500	[ "31", "304", "4230", "43880", "547338", "6924960", "94714620", "1375878816", "21273204330", "348919244768", "6056244249682", "110955673493568", "2140465858763844", "43379533256972640", "921616584567907176", "20485188316420940640", "475499882089797554181", "11506280235885243825696" ]	Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-7 places.

A000401

[ "0", "1", "2", "3", "4", "5", "6", "7", "8", "9", "10", "11", "12", "13", "15", "16", "17", "18", "19", "20", "21", "22", "23", "24", "25", "26", "27", "28", "29", "31", "32", "33", "34", "35", "36", "37", "38", "39", "40", "41", "42", "43", "44", "45", "47", "48", "49", "50", "51", "52", "53", "54", "55", "57", "58", "59", "60", "61", "63", "64", "65", "66", "67", "68", "69", "70", "71", "72", "73", "74", "75", "76" ]

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

A000402

[ "0", "0", "1", "6", "41", "293", "2309", "19975", "189524", "1960041", "21993884", "266361634", "3465832370", "48245601976", "715756932697", "11277786883720", "188135296651083", "3313338641692957", "61444453534759589", "1196988740015236617", "24442368179977776766", "522124104504306695929" ]

Number of permutations of [n] in which the longest increasing run has length 3.

A000403

[ "1", "1", "2", "2", "5", "6" ]

Number of simple equifacetted 3-manifolds with n faces.

A000404

[ "2", "5", "8", "10", "13", "17", "18", "20", "25", "26", "29", "32", "34", "37", "40", "41", "45", "50", "52", "53", "58", "61", "65", "68", "72", "73", "74", "80", "82", "85", "89", "90", "97", "98", "100", "101", "104", "106", "109", "113", "116", "117", "122", "125", "128", "130", "136", "137", "145", "146", "148", "149", "153", "157", "160", "162", "164", "169", "170", "173", "178" ]

Numbers that are the sum of 2 nonzero squares.

A000405

[ "1", "1", "6", "51", "561", "7556", "120196", "2201856", "45592666", "1051951026", "26740775306", "742069051906", "22310563733864", "722108667742546", "25024187820786357", "924161461265888370", "36223781285638309482", "1501552062016443881514" ]

Number of 6-level labeled rooted trees with n leaves.

A000406

[ "1", "6", "57", "741", "12244", "245755", "5809875", "158198200", "4877852505", "168055077875", "6400217406500", "267058149580823", "12118701719205803", "594291742526530761", "31323687504696772151", "1766116437541895988303", "106080070002238888908150" ]

Coefficients of iterated exponentials.

A000407

[ "1", "6", "60", "840", "15120", "332640", "8648640", "259459200", "8821612800", "335221286400", "14079294028800", "647647525324800", "32382376266240000", "1748648318376960000", "101421602465863680000", "6288139352883548160000", "415017197290314178560000" ]

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

A000408

[ "3", "6", "9", "11", "12", "14", "17", "18", "19", "21", "22", "24", "26", "27", "29", "30", "33", "34", "35", "36", "38", "41", "42", "43", "44", "45", "46", "48", "49", "50", "51", "53", "54", "56", "57", "59", "61", "62", "65", "66", "67", "68", "69", "70", "72", "73", "74", "75", "76", "77", "78", "81", "82", "83", "84", "86", "88", "89", "90", "91", "93", "94", "96", "97", "98", "99", "101", "102", "104" ]

Numbers that are the sum of three nonzero squares.

A000409

[ "0", "6", "350", "43260", "14591171", "14657461469", "46173502811223", "474928141312623525", "16489412944755088235117", "1985178211854071817861662307", "846428472480689964807653763864449", "1299141117072945982773752362381072143359", "7268140170419155675761326840423792818571154945", "149650282980396792665043455999899697765782372693740287" ]

Singular n X n (0,1)-matrices: the number of n X n (0,1)-matrices having distinct, nonzero ordered rows, but having at least two equal columns or at least one zero column.

A000410

[ "0", "0", "6", "425", "65625", "27894671", "35716401889", "144866174953833" ]

Number of singular n X n rational (0,1)-matrices.

A000411

[ "6", "522", "152166", "93241002", "97949265606", "157201459863882", "357802951084619046", "1096291279711115037162", "4350684698032741048452486", "21709332137467778453687752842", "133032729004732721625426681085926", "982136301747914281420205946546842922", "8597768767880274820173388403096814519366" ]

Generalized tangent numbers d(6,n).

A000412

[ "3", "7", "16", "31", "57", "97", "162", "257", "401", "608", "907", "1325", "1914", "2719", "3824", "5313", "7316", "9973", "13495", "18105", "24132", "31938", "42021", "54948", "71484", "92492", "119120", "152686", "194887", "247693", "313613", "395547", "497154", "622688", "777424", "967525", "1200572", "1485393", "1832779", "2255317" ]

Number of bipartite partitions of n white objects and 3 black ones.

A000413

[ "1", "7", "19", "57", "81", "251", "437", "691", "739", "1743", "3695", "6619", "8217", "9771", "14771", "15155", "16831", "18805", "26745", "30551", "41755", "46297", "54339", "72359", "86407", "96969", "131059", "344859", "395231", "519963", "607141", "677397", "741509", "893019", "917217", "1288415", "1406811", "1789599", "1827927", "3085785", "3216051", "3444439", "3524869" ]

Let A(n) = #{(i,j,k): i^2 + j^2 + k^2 <= n}, V(n) = (4/3)*Pi*n^(3/2), P(n) = A(n) - V(n); A000092 gives values of n where |P(n)| sets a new record; sequence gives A(A000092(n)).

A000414

[ "4", "7", "10", "12", "13", "15", "16", "18", "19", "20", "21", "22", "23", "25", "26", "27", "28", "30", "31", "33", "34", "35", "36", "37", "38", "39", "40", "42", "43", "44", "45", "46", "47", "48", "49", "50", "51", "52", "53", "54", "55", "57", "58", "59", "60", "61", "62", "63", "64", "65", "66", "67", "68", "69", "70", "71", "72", "73", "74", "75", "76", "77", "78", "79", "80", "81" ]

Numbers that are the sum of 4 nonzero squares.

A000415

[ "2", "5", "8", "10", "13", "17", "18", "20", "26", "29", "32", "34", "37", "40", "41", "45", "50", "52", "53", "58", "61", "65", "68", "72", "73", "74", "80", "82", "85", "89", "90", "97", "98", "101", "104", "106", "109", "113", "116", "117", "122", "125", "128", "130", "136", "137", "145", "146", "148", "149", "153", "157", "160", "162", "164", "170", "173", "178", "180", "181" ]

Numbers that are the sum of 2 but no fewer nonzero squares.

A000416

[ "1", "7", "28", "105", "357", "1197", "3857", "12300", "38430", "118874", "362670", "1095430", "3271751", "9673993", "28310881", "82033609", "235359901", "668779076", "1882412994", "5249817573", "14510628853" ]

Number of 6-dimensional partitions of n.

A000417

[ "1", "7", "28", "105", "357", "1232", "4067", "13301", "42357", "132845", "409262", "1243767", "3727360", "11036649", "32300795", "93538278", "268164868", "761656685", "2144259516", "5986658951", "16583102077", "45593269265", "124464561544", "337479729179", "909156910290", "2434121462871", "6478440788169" ]

Euler transform of A000389.

A000418

[ "0", "0", "0", "0", "0", "0", "0", "1", "7", "34", "136", "487", "1615", "5079", "15349", "45009", "128899", "362266", "1002681", "2740448", "7411408", "19865445", "52840977", "139624510", "366803313", "958696860", "2494322662", "6463281890", "16686206047", "42935345688", "110142163940", "281763465941" ]

Number of n-node rooted trees of height 7.

A000419

[ "3", "6", "11", "12", "14", "19", "21", "22", "24", "27", "30", "33", "35", "38", "42", "43", "44", "46", "48", "51", "54", "56", "57", "59", "62", "66", "67", "69", "70", "75", "76", "77", "78", "83", "84", "86", "88", "91", "93", "94", "96", "99", "102", "105", "107", "108", "110", "114", "115", "118", "120", "123", "126", "129", "131", "132", "133", "134", "138", "139", "140", "141", "142" ]

Numbers that are the sum of 3 but no fewer nonzero squares.

A000420

[ "1", "7", "49", "343", "2401", "16807", "117649", "823543", "5764801", "40353607", "282475249", "1977326743", "13841287201", "96889010407", "678223072849", "4747561509943", "33232930569601", "232630513987207", "1628413597910449", "11398895185373143", "79792266297612001", "558545864083284007" ]

Powers of 7: a(n) = 7^n.

A000421

[ "1", "2", "6", "20", "91", "509", "3608", "31856", "340416", "4269971", "61133757", "978098997", "17228295555", "330552900516", "6853905618223", "152626436936272", "3631575281503404", "91928898608055819", "2466448432564961852", "69907637101781318907" ]

Number of isomorphism classes of connected 3-regular (trivalent, cubic) loopless multigraphs of order 2n.

A000422

[ "1", "21", "321", "4321", "54321", "654321", "7654321", "87654321", "987654321", "10987654321", "1110987654321", "121110987654321", "13121110987654321", "1413121110987654321", "151413121110987654321", "16151413121110987654321", "1716151413121110987654321", "181716151413121110987654321" ]

Concatenation of numbers from n down to 1.

A000423

[ "2", "3", "6", "12", "18", "24", "36", "48", "54", "72", "96", "108", "144", "162", "192", "216", "288", "324", "384", "432", "486", "576", "648", "768", "864", "972", "1152", "1296", "1458", "1536", "1728", "1944", "2304", "2592", "2916", "3072", "3456", "3888", "4374", "4608", "5184", "5832", "6144", "6912", "7776", "8748", "9216", "10368", "11664" ]

a(n) is smallest number > a(n-1) of form a(i)*a(j), i < j < n.

A000424

[ "7", "85", "1660", "48076", "1942416", "104587344", "7245893376", "628308907776", "66687811660800", "8506654697548800", "1284292319599411200", "226530955276874956800", "46165213716463676620800", "10765453901922078105600000", "2848453606917036402278400000", "848800150518516674081587200000" ]

Differences of reciprocals of unity.

A000425

[ "2", "0", "0", "8", "30", "192", "1344", "10800", "97434", "976000", "10749024", "129103992", "1679495350", "23525384064", "353028802560", "5650370001120", "96082828074162", "1729886440780800", "32874134679574208", "657589108734075240", "13811277748363437006", "303884178002526338624" ]

Coefficients of ménage hit polynomials.

A000426

[ "0", "1", "1", "1", "8", "35", "211", "1459", "11584", "103605", "1030805", "11291237", "135015896", "1749915271", "24435107047", "365696282855", "5839492221440", "99096354764009", "1780930394412009", "33789956266629001", "674939337282352360", "14157377139256183723", "311135096550816014651" ]

Coefficients of ménage hit polynomials.

A000427

[ "1", "8", "36", "148", "554", "2024", "7134", "24796", "84625", "285784", "953430", "3151332", "10314257", "33457972", "107557792", "342732670", "1082509680", "3389190112", "10518508294", "32361863632" ]

Number of 7-dimensional partitions of n.

A000428

[ "1", "8", "36", "148", "554", "2094", "7624", "27428", "96231", "332159", "1126792", "3769418", "12437966", "40544836", "130643734", "416494314", "1314512589", "4110009734", "12737116845", "39144344587", "119350793207", "361173596536", "1085171968872" ]

Euler transform of A000579.

A000429

[ "0", "0", "0", "0", "0", "0", "0", "0", "1", "8", "43", "188", "728", "2593", "8706", "27961", "86802", "262348", "776126", "2256418", "6466614", "18311915", "51334232", "142673720", "393611872", "1078955836", "2941029334", "7977065816", "21541492856", "57942770689", "155304829763", "414934057486" ]

Number of n-node rooted trees of height 8.

A000430

[ "2", "3", "4", "5", "7", "9", "11", "13", "17", "19", "23", "25", "29", "31", "37", "41", "43", "47", "49", "53", "59", "61", "67", "71", "73", "79", "83", "89", "97", "101", "103", "107", "109", "113", "121", "127", "131", "137", "139", "149", "151", "157", "163", "167", "169", "173", "179", "181", "191", "193", "197", "199", "211", "223" ]

Primes and squares of primes.

A000431

[ "0", "0", "0", "2", "16", "88", "416", "1824", "7680", "31616", "128512", "518656", "2084864", "8361984", "33497088", "134094848", "536608768", "2146926592", "8588754944", "34357248000", "137433710592", "549744803840", "2199000186880", "8796044787712", "35184271425536", "140737278640128", "562949517213696" ]

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

A000432

[ "8", "52", "288", "1424", "6648", "29700", "128800", "545600", "2269672", "9303140", "37672216", "150998016", "599988696", "2366216164", "9270987656", "36116062832", "139978757920", "540069059028", "2075217121688", "7944690769952", "30313624200640", "115312027433188", "437420730644304", "1655047867097280", "6247339311097296", "23530440547115428", "88447214709073696", "331832490378209152", "1242766581420901656", "4646714574562484628", "17347357264162110368", "64668460220964604944", "240747014238189337840", "895102104022837748484", "3323982608759454833032", "12329573838525875316560", "45684294664598118867184", "169098457957523787786644" ]

Series-parallel numbers.

A000433

[ "0", "1", "2", "3", "4", "5", "6", "7", "10", "11", "12", "13", "14", "15", "16", "17", "20", "21", "22", "23", "24", "25", "26", "27", "30", "31", "32", "100", "101", "102", "103", "104", "105", "106", "107", "110", "111", "112", "113", "114", "115", "116", "117", "120", "121", "122", "123", "124", "125", "126", "127", "130", "131", "132", "200", "201", "202", "203" ]

n written in base where place values are positive cubes.

A000434

[ "0", "0", "0", "1", "8", "67", "602", "5811", "60875", "690729", "8457285", "111323149", "1569068565", "23592426102", "377105857043", "6387313185576", "114303481217657", "2155348564847332", "42719058006864690", "887953677898186108", "19316200230609433690", "438920223893512987430" ]

Number of permutations of [n] in which the longest increasing run has length 4.

A000435

[ "0", "1", "8", "78", "944", "13800", "237432", "4708144", "105822432", "2660215680", "73983185000", "2255828154624", "74841555118992", "2684366717713408", "103512489775594200", "4270718991667353600", "187728592242564421568", "8759085548690928992256", "432357188322752488126152", "22510748754252398927872000" ]

Normalized total height of all nodes in all rooted trees with n labeled nodes.

A000436

[ "1", "8", "352", "38528", "7869952", "2583554048", "1243925143552", "825787662368768", "722906928498737152", "806875574817679474688", "1118389087843083461066752", "1884680130335630169428983808", "3794717805092151129643367268352" ]

Generalized Euler numbers c(3,n).

A000437

[ "0", "9", "41", "81", "146", "194", "306", "369", "425", "594", "689", "866", "1109", "1161", "1154", "1361", "1634", "1781", "1889", "2141", "2729", "2609", "3626", "3366", "3566", "3449", "3506", "4241", "4289", "4826", "5066", "5381", "7034", "5561", "6254", "7229", "7829", "8186", "8069", "8126", "8609", "8921", "8774" ]

Smallest nonnegative number that is the sum of 3 squares in exactly n ways.

A000438

[ "1", "1", "6", "6240", "1225566720", "252282619805368320", "98758655816833727741338583040" ]

Number of 1-factorizations of complete graph K_{2n}.

A000439

[ "9", "30", "69", "133", "230", "369", "560", "814", "1143", "1560", "2079", "2715", "3484", "4403", "5490", "6764", "8245", "9954", "11913", "14145", "16674", "19525", "22724", "26298", "30275", "34684", "39555", "44919", "50808", "57255", "64294", "71960", "80289", "89318", "99085", "109629", "120990", "133209", "146328", "160390", "175439", "191520", "208679" ]

Powers of rooted tree enumerator.

A000440

[ "9", "30", "180", "980", "8326", "70272", "695690", "7518720", "89193276", "1148241458", "15947668065", "237613988040", "3780133322620", "63945806121448", "1146081593303784", "21693271558730304", "432411684714253605", "9053476937543082240", "198641103956454088919" ]

Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-4 places.

A000441

[ "0", "1", "9", "34", "95", "210", "406", "740", "1161", "1920", "2695", "4116", "5369", "7868", "9690", "13640", "16116", "22419", "25365", "34160", "38640", "50622", "55154", "73320", "77225", "100100", "107730", "135576", "141085", "182340", "184760", "233616", "243408", "297738", "301420", "385110", "377511", "467210", "478842" ]

a(n) = Sum_{k=1..n-1} k*sigma(k)*sigma(n-k).

A000442

[ "1", "1", "8", "216", "13824", "1728000", "373248000", "128024064000", "65548320768000", "47784725839872000", "47784725839872000000", "63601470092869632000000", "109903340320478724096000000", "241457638684091756838912000000", "662559760549147780765974528000000", "2236139191853373760085164032000000000" ]

a(n) = (n!)^3.

A000443

[ "325", "425", "625", "650", "725", "845", "850", "925", "1025", "1250", "1300", "1325", "1445", "1450", "1525", "1690", "1700", "1825", "1850", "2050", "2225", "2425", "2500", "2525", "2600", "2650", "2725", "2825", "2873", "2890", "2900", "2925", "3050", "3125", "3380", "3400", "3425", "3650", "3700", "3725", "3757", "3825", "3925", "4100" ]

Numbers that are the sum of 2 squares in exactly 3 ways.

A000444

[ "9", "64", "326", "1433", "5799", "22224", "81987", "293987", "1031298", "3555085", "12081775", "40576240", "134919788", "444805274", "1455645411", "4733022100", "15302145060", "49223709597", "157629612076", "502736717207", "1597541346522", "5059625685739", "15975936032821", "50304490599602" ]

Number of partially labeled rooted trees with n nodes (3 of which are labeled).

A000445

[ "9", "77", "1224", "7888", "202124", "1649375" ]

Latest possible occurrence of the first consecutive pair of n-th power residues, modulo any prime.

A000446

[ "0", "25", "325", "1105", "4225", "5525", "203125", "27625", "71825", "138125", "2640625", "160225", "17850625", "1221025", "1795625", "801125", "1650390625", "2082925", "49591064453125", "4005625", "44890625", "2158203125", "30525625", "5928325", "303460625", "53955078125" ]

Smallest number that is the sum of 2 squares (allowing zeros) in exactly n ways.

A000447

[ "0", "1", "10", "35", "84", "165", "286", "455", "680", "969", "1330", "1771", "2300", "2925", "3654", "4495", "5456", "6545", "7770", "9139", "10660", "12341", "14190", "16215", "18424", "20825", "23426", "26235", "29260", "32509", "35990", "39711", "43680", "47905", "52394", "57155", "62196", "67525", "73150", "79079", "85320", "91881", "98770", "105995", "113564", "121485" ]

a(n) = 1^2 + 3^2 + 5^2 + 7^2 + ... + (2*n-1)^2 = n*(4*n^2 - 1)/3.

A000448

[ "0", "25", "325", "1105", "4225", "5525", "27625", "27625", "71825", "138125", "160225", "160225", "801125", "801125", "801125", "801125", "2082925", "2082925", "4005625", "4005625", "5928325", "5928325", "5928325", "5928325", "29641625", "29641625", "29641625", "29641625", "29641625", "29641625", "29641625", "29641625" ]

Smallest number that is the sum of 2 squares in at least n ways.

A000449

[ "1", "0", "10", "40", "315", "2464", "22260", "222480", "2447445", "29369120", "381798846", "5345183480", "80177752655", "1282844041920", "21808348713320", "392550276838944", "7458455259940905", "149169105198816960", "3132551209175157490", "68916126601853463240" ]

Rencontres numbers: number of permutations of [n] with exactly 3 fixed points.

A000450

[ "1", "10", "50", "385", "3130", "28764", "291900", "3249210", "39367395", "515874470", "7270929806", "109691447395", "1763782644020", "30114243100760", "544123405603800", "10373304279494964", "208092476689208805", "4381794437494059810", "96635643877147885450", "2227562535558247510885" ]

Coefficients of ménage hit polynomials.

A000451

[ "0", "9", "41", "81", "146", "194", "306", "369", "425", "594", "689", "866", "1109", "1154", "1154", "1361", "1634", "1781", "1889", "2141", "2609", "2609", "3366", "3366", "3449", "3449", "3506", "4241", "4289", "4826", "5066", "5381", "5561", "5561", "6254", "7229", "7829", "8069", "8069", "8126", "8609", "8774", "8774" ]

Smallest number that is the sum of 3 squares in at least n ways.

A000452

[ "1", "2", "3", "5", "6", "7", "8", "10", "11", "13", "14", "15", "16", "17", "19", "21", "22", "23", "24", "26", "27", "29", "30", "31", "33", "34", "35", "37", "38", "39", "40", "41", "42", "43", "46", "47", "48", "51", "53", "54", "55", "56", "57", "58", "59", "61", "62", "65", "66", "67", "69", "70", "71", "73", "74", "77", "78", "79", "80", "81", "82", "83", "85", "86", "87", "88", "89", "91", "93", "94", "95", "97", "101" ]

The greedy sequence of integers which avoids 3-term geometric progressions.

A000453

[ "1", "10", "65", "350", "1701", "7770", "34105", "145750", "611501", "2532530", "10391745", "42355950", "171798901", "694337290", "2798806985", "11259666950", "45232115901", "181509070050", "727778623825", "2916342574750", "11681056634501", "46771289738810", "187226356946265", "749329038535350" ]

Stirling numbers of the second kind, S(n,4).

A000454

[ "1", "10", "85", "735", "6769", "67284", "723680", "8409500", "105258076", "1414014888", "20313753096", "310989260400", "5056995703824", "87077748875904", "1583313975727488", "30321254007719424", "610116075740491776" ]

Unsigned Stirling numbers of first kind s(n,4).

A000455

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

Digits of powers of 2.

A000456

[ "0", "0", "0", "0", "1", "10", "99", "1024", "11304", "133669", "1695429", "23023811", "333840443", "5153118154", "84426592621", "1463941342191", "26793750988542", "516319125748337", "10451197169218523", "221738082618710329", "4921234092461339819", "114041894068935641488" ]

Number of permutations of [n] in which the longest increasing run has length 5.

A000457

[ "1", "10", "105", "1260", "17325", "270270", "4729725", "91891800", "1964187225", "45831035250", "1159525191825", "31623414322500", "924984868933125", "28887988983603750", "959493919812553125", "33774185977401870000", "1255977541034632040625" ]

Exponential generating function: (1+3*x)/(1-2*x)^(7/2).

A000458

[ "1", "1", "2", "4", "10", "25", "49", "88", "237", "500", "1412", "3570", "12846", "36072", "126504", "493920", "2358720", "12292224", "49984224", "171237888", "642804078", "1980490350", "6380883000", "27032104440", "117961599600", "555861355920", "2623445776224" ]

a(0) = a(1) = 1; thereafter a(n) = sigma(a(n-1)) + sigma(a(n-2)).

A000459

[ "1", "0", "1", "10", "297", "13756", "925705", "85394646", "10351036465", "1596005408152", "305104214112561", "70830194649795010", "19629681235869138841", "6401745422388206166420", "2427004973632598297444857", "1058435896607583305978409166", "526149167104704966948064477665" ]

Number of multiset permutations of {1, 1, 2, 2, ..., n, n} with no fixed points.

A000460

[ "1", "11", "66", "302", "1191", "4293", "14608", "47840", "152637", "478271", "1479726", "4537314", "13824739", "41932745", "126781020", "382439924", "1151775897", "3464764515", "10414216090", "31284590870", "93941852511", "282010106381", "846416194536", "2540053889352", "7621839388981", "22869007827143" ]

Eulerian numbers (Euler's triangle: column k=3 of A008292, column k=2 of A173018).

A000461

[ "1", "22", "333", "4444", "55555", "666666", "7777777", "88888888", "999999999", "10101010101010101010", "1111111111111111111111", "121212121212121212121212", "13131313131313131313131313", "1414141414141414141414141414", "151515151515151515151515151515", "16161616161616161616161616161616" ]

Concatenate n n times.

A000462

[ "1", "2", "10", "11", "12", "100", "101", "102", "110", "1000", "1001", "1002", "1010", "1011", "10000", "10001", "10002", "10010", "10011", "10012", "100000", "100001", "100002", "100010", "100011", "100012", "100100", "1000000", "1000001", "1000002", "1000010", "1000011", "1000012", "1000100", "1000101", "10000000", "10000001" ]

Numbers written in base of triangular numbers.

A000463

[ "1", "1", "2", "4", "3", "9", "4", "16", "5", "25", "6", "36", "7", "49", "8", "64", "9", "81", "10", "100", "11", "121", "12", "144", "13", "169", "14", "196", "15", "225", "16", "256", "17", "289", "18", "324", "19", "361", "20", "400", "21", "441", "22", "484", "23", "529", "24", "576", "25", "625", "26", "676", "27", "729", "28", "784", "29", "841", "30", "900", "31", "961", "32", "1024", "33", "1089", "34", "1156", "35", "1225", "36", "1296" ]

n followed by n^2.

A000464

[ "1", "11", "361", "24611", "2873041", "512343611", "129570724921", "44110959165011", "19450718635716001", "10784052561125704811", "7342627959965776406281", "6023130568334172003579011", "5858598896811701995459355761", "6667317162352419006959182803611", "8776621742176931117228228227924441" ]

Expansion of sin x /cos 2x.

A000465

[ "5", "12", "29", "57", "109", "189", "323", "522", "831", "1279", "1941", "2876", "4215", "6066", "8644", "12151", "16933", "23336", "31921", "43264", "58250", "77825", "103362", "136371", "178975", "233532", "303268", "391831", "504069", "645520", "823419", "1046067", "1324136", "1669950", "2099104", "2629685", "3284325", "4089300" ]

Number of bipartite partitions of n white objects and 4 black ones.

A000466

[ "-1", "3", "15", "35", "63", "99", "143", "195", "255", "323", "399", "483", "575", "675", "783", "899", "1023", "1155", "1295", "1443", "1599", "1763", "1935", "2115", "2303", "2499", "2703", "2915", "3135", "3363", "3599", "3843", "4095", "4355", "4623", "4899", "5183", "5475", "5775", "6083", "6399", "6723", "7055", "7395" ]

a(n) = 4*n^2 - 1.

A000467

[ "0", "0", "0", "0", "0", "1", "12", "137", "1602", "19710", "257400", "3574957", "52785901", "827242933", "13730434111", "240806565782", "4452251786946", "86585391630673", "1767406549387381", "37790452850585180", "844817788372455779", "19711244788916894489", "479203883157602851294" ]

Number of permutations of [n] in which the longest increasing run has length 6.

A000468

[ "1", "12", "144", "1750", "23420", "303240", "3641100", "46113200", "575360400", "7346545000", "112402762000", "1351035564000", "16432451210000", "221411634520000", "2657142036440000", "34327724461500000", "434157115760200000", "5432127413542400000", "67405553164731000000", "1053071060221172000000" ]

Powers of ten written in base 8.

A000469

[ "1", "6", "10", "14", "15", "21", "22", "26", "30", "33", "34", "35", "38", "39", "42", "46", "51", "55", "57", "58", "62", "65", "66", "69", "70", "74", "77", "78", "82", "85", "86", "87", "91", "93", "94", "95", "102", "105", "106", "110", "111", "114", "115", "118", "119", "122", "123", "129", "130", "133", "134", "138", "141", "142", "143", "145", "146", "154", "155", "158" ]

1 together with products of 2 or more distinct primes.

A000470

[ "13", "72", "595", "4096", "39078", "379760", "4181826", "49916448", "647070333", "9035216428", "135236990388", "2159812592384", "36658601139066", "658942295734944", "12504663617290908", "249823152134646144", "5241223014084306270", "115206851288747267148", "2647678812396326064043" ]

Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-5 places.

A000471

[ "0", "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(sinh(n)).

A000472

[ "2", "5", "28", "802", "643726", "414383582242", "171713753231982206218246", "29485613049014079571725771288849499850026859242" ]

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

A000473

[ "14", "386", "5868", "65954", "614404", "5030004", "37460376", "259477218", "1697186964", "10596579708", "63663115880", "370293754740", "2095108370600", "11574690111400", "62629794691632", "332742342741090", "1739371969822260", "8961709528660140", "45576855706440520", "229087231033907708" ]

Number of genus 0 rooted maps with 5 faces and n vertices.

A000474

[ "1", "1", "1", "6", "396", "526915620", "1132835421602062347" ]

Number of nonisomorphic 1-factorizations of complete graph K_{2n}.

A000475

[ "1", "0", "15", "70", "630", "5544", "55650", "611820", "7342335", "95449640", "1336295961", "20044438050", "320711010620", "5452087178160", "98137569209940", "1864613814984984", "37292276299704525", "783137802293789040", "17229031650463366195", "396267727960657413630" ]

Rencontres numbers: number of permutations of [n] with exactly 4 fixed points.

A000476

[ "15", "72", "609", "4960", "46188", "471660", "5275941", "64146768", "842803767", "11902900380", "179857257960", "2895705788736", "49491631601635", "895010868095256", "17074867330880805", "342733960299356800", "7220616209235766260", "159312370008282356844", "3673720238903201471593" ]

Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-1 places.

A000477

[ "0", "1", "15", "76", "275", "720", "1666", "3440", "6129", "11250", "17545", "28896", "41405", "65072", "85950", "128960", "162996", "238545", "286995", "404600", "482160", "662112", "756470", "1042560", "1150625", "1549730", "1732590", "2257920", "2443105", "3250800", "3421160", "4452096", "4791600", "6039522", "6296500" ]

a(n) = Sum_{k=1..n-1} k^2*sigma(k)*sigma(n-k).

A000478

[ "15", "105", "490", "1918", "6825", "22935", "74316", "235092", "731731", "2252341", "6879678", "20900922", "63259533", "190957923", "575363776", "1731333808", "5205011031", "15638101281", "46962537810", "140988276150", "423174543025", "1269959836015", "3810785476980", "11434235478348", "34306598748315", "102927849307725" ]

Number of ways of placing n labeled balls into 3 indistinguishable boxes with at least 2 balls in each box.

A000479

[ "1", "1", "1", "2", "24", "1344", "1128960", "12198297600", "2697818265354240", "15224734061278915461120", "2750892211809148994633229926400", "19464657391668924966616671344752852992000" ]

Number of 1-factorizations of K_{n,n}.

A000480

[ "1", "0", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1", "0", "0", "0", "0", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1", "0", "0", "0", "0", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1" ]

a(n) = floor(cos(n)).

A000481

[ "1", "15", "140", "1050", "6951", "42525", "246730", "1379400", "7508501", "40075035", "210766920", "1096190550", "5652751651", "28958095545", "147589284710", "749206090500", "3791262568401", "19137821912055", "96416888184100", "485000783495250", "2436684974110751", "12230196160292565", "61338207158409090" ]

Stirling numbers of the second kind, S(n,5).

A000482

[ "1", "15", "175", "1960", "22449", "269325", "3416930", "45995730", "657206836", "9957703756", "159721605680", "2706813345600", "48366009233424", "909299905844112", "17950712280921504", "371384787345228000", "8037811822645051776", "181664979520697076096", "4280722865357147142912", "105005310755917452984576" ]

Unsigned Stirling numbers of first kind s(n,5).

A000483

[ "15", "210", "2380", "26432", "303660", "3678840", "47324376", "647536032", "9418945536", "145410580224", "2377609752960", "41082721413120", "748459539843840", "14345340443665920", "288650580508961280", "6085390148673177600", "134167064248901376000", "3088040233895705088000", "74077507611407752704000", "1849221425299053367296000" ]

Associated Stirling numbers: second order reciprocal Stirling numbers (Fekete) [[n, 3]]. The number of 3-orbit permutations of an n-set with at least 2 elements in each orbit.

A000484

[ "1", "1", "0", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "0", "1", "1", "1", "0", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "0", "1", "1", "0" ]

a(n) = round(cos(n)).

A000485

[ "16", "125", "680", "3135", "13155", "51873", "195821", "715614", "2550577", "8911942", "30640888", "103951415", "348724844", "1158722880", "3818514232", "12493703403", "40620949971", "131336770375", "422536529249", "1353341880777", "4317248276746", "13722302173753" ]

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

A000486

[ "16", "150", "926", "4788", "22548", "100530", "433162", "1825296", "7577120", "31130190", "126969558", "515183724", "2082553132", "8395437930", "33776903714", "135691891272", "544517772984", "2183315948550", "8748985781230", "35043081823140", "140313684667076" ]

One half of the number of permutations of [n] such that the differences have 4 runs with the same signs.

A000487

[ "16", "272", "2880", "24576", "185856", "1304832", "8728576", "56520704", "357888000", "2230947840", "13754155008", "84134068224", "511780323328", "3100738912256", "18733264797696", "112949304754176", "680032201605120", "4090088616099840", "24582312700149760", "147669797096652800" ]

Number of permutations of length n with exactly two valleys.

A000488

[ "16", "361", "3362", "16384", "55744", "152166", "355688", "739328", "1415232", "2529614", "4261454", "6885376", "10708160", "16054580", "23494584", "33554432", "46698624", "64037790", "86342918", "114163712", "149518720", "193356526", "246232840", "311635968", "390600000" ]

Generalized tangent numbers d_(n,3).

A000489

[ "1", "16", "435", "7136", "99350", "1234032", "14219212", "155251840", "1628202762", "16550991200", "164111079110", "1594594348800", "15235525651840", "143518352447680", "1335670583147400", "12301278983461376", "112264111607438906", "1016361486936571680", "9136254276320346046" ]

Card matching: Coefficients B[n,3] of t^3 in the reduced hit polynomial A[n,n,n](t).

A000490

[ "1", "16", "1280", "249856", "90767360", "52975108096", "45344872202240", "53515555843342336", "83285910482761809920", "165262072909347030040576", "407227428060372417275494400", "1219998300294918683087199010816", "4366953142363907901751614431559680", "18406538229888710811704852978971181056" ]

Generalized Euler numbers c(4,n).

A000491

[ "7", "19", "47", "97", "189", "339", "589", "975", "1576", "2472", "3804", "5727", "8498", "12400", "17874", "25433", "35818", "49908", "68939", "94378", "128234", "172917", "231630", "308240", "407804", "536412", "701910", "913773", "1184022", "1527165", "1961432", "2508762", "3196473", "4057403", "5132066" ]

Number of bipartite partitions of n white objects and 5 black ones.

A000492

[ "20", "154", "1676", "14292", "155690", "1731708", "21264624", "280260864", "3970116255", "60113625680", "969368687752", "16588175089420", "300272980075896", "5733025551810600", "115148956467702600", "2427199940533198992", "53576182138937428377", "1235917889588345408586" ]

Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-6 places.

A000493

[ "0", "0", "0", "0", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1", "0", "0", "0", "0", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1", "0", "0", "0", "-1", "-1", "-1", "0", "0" ]

a(n) = floor(sin(n)).

A000494

[ "0", "1", "1", "0", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "0", "1", "1", "1", "0", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "0", "1", "1", "1", "0", "-1", "-1", "0", "1", "1", "0", "-1", "-1", "0", "1", "1", "0", "-1" ]

Nearest integer to sin(n).

A000495

[ "0", "1", "4", "10", "27", "74", "202", "548", "1490", "4052", "11013", "29937", "81377", "221207", "601302", "1634509", "4443055", "12077476", "32829985", "89241150", "242582598", "659407867", "1792456423", "4872401723", "13244561065", "36002449669", "97864804714", "266024120301", "723128532146", "1965667148572" ]

Nearest integer to sinh(n).

A000496

[ "1", "1", "2", "6", "24", "44", "80", "144", "260", "476", "872", "1600", "2940", "5404", "9936", "18272", "33604", "61804", "113672", "209072", "384540", "707276", "1300880", "2392688", "4400836", "8094396", "14887912", "27383136", "50365436", "92636476", "170385040", "313386944", "576408452", "1060180428", "1949975816", "3586564688" ]

Restricted permutations.

A000497

[ "1", "25", "490", "9450", "190575", "4099095", "94594500", "2343240900", "62199262125", "1764494857125", "53338158823950", "1712934942468750", "58274046742786875", "2094379201311271875", "79318164037837725000", "3157886388887074845000" ]

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

A000498

[ "1", "26", "302", "2416", "15619", "88234", "455192", "2203488", "10187685", "45533450", "198410786", "848090912", "3572085255", "14875399450", "61403313100", "251732291184", "1026509354985", "4168403181210", "16871482830550", "68111623139600" ]

Eulerian numbers (Euler's triangle: column k=4 of A008292, column k=3 of A173018)

A000499

[ "0", "1", "27", "184", "875", "2700", "7546", "17600", "35721", "72750", "126445", "223776", "353717", "595448", "843750", "1349120", "1827636", "2808837", "3600975", "5306000", "6667920", "9599172", "11509982", "16416000", "19015625", "26605670", "30902310", "41686848", "46948825", "64233000", "70306760", "94089216" ]

a(n) = Sum_{k=1..n-1} k^3*sigma(k)*sigma(n-k).

A000500

[ "31", "304", "4230", "43880", "547338", "6924960", "94714620", "1375878816", "21273204330", "348919244768", "6056244249682", "110955673493568", "2140465858763844", "43379533256972640", "921616584567907176", "20485188316420940640", "475499882089797554181", "11506280235885243825696" ]

Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-7 places.