cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 11 results. Next

A367057 G.f. satisfies A(x) = 1 + x*A(x)^3 + x^3*A(x).

Original entry on oeis.org

1, 1, 3, 13, 59, 294, 1549, 8477, 47715, 274468, 1606284, 9533595, 57247969, 347169053, 2123148153, 13079296531, 81087402683, 505543820304, 3167578950478, 19935616736595, 125971005957924, 798883392476824, 5083047458454395, 32439034490697090
Offset: 0

Views

Author

Seiichi Manyama, Nov 04 2023

Keywords

Crossrefs

Cf. A366676, A367058, A367059, A367060, A367061, A367062.

Programs

  • PARI
    a(n) = sum(k=0, n\3, binomial(2*n-5*k+1, k)*binomial(3*n-8*k, n-3*k)/(2*n-5*k+1));

Formula

a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-5*k+1,k) * binomial(3*n-8*k,n-3*k)/(2*n-5*k+1).

A367058 G.f. satisfies A(x) = 1 + x*A(x)^3 + x^3*A(x)^2.

Original entry on oeis.org

1, 1, 3, 13, 60, 301, 1595, 8774, 49631, 286870, 1686876, 10059301, 60689041, 369762262, 2271892435, 14060917955, 87579290486, 548558815484, 3453077437532, 21833406999880, 138603490377008, 883075187803622, 5644796991703781, 36191055027026410
Offset: 0

Views

Author

Seiichi Manyama, Nov 04 2023

Keywords

Crossrefs

Cf. A366676, A367057, A367059, A367060, A367061, A367062.

Programs

  • PARI
    a(n) = sum(k=0, n\3, binomial(2*n-4*k+1, k)*binomial(3*n-7*k, n-3*k)/(2*n-4*k+1));

Formula

a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-4*k+1,k) * binomial(3*n-7*k,n-3*k)/(2*n-4*k+1).

A367059 G.f. satisfies A(x) = 1 + x*A(x)^3 + x^3*A(x)^3.

Original entry on oeis.org

1, 1, 3, 13, 61, 309, 1651, 9153, 52161, 303681, 1798459, 10800237, 65614237, 402544597, 2490398139, 15519350593, 97326638145, 613786324353, 3890080513395, 24764386415821, 158281551244029, 1015314894877237, 6534249237530115, 42178452056044929
Offset: 0

Views

Author

Seiichi Manyama, Nov 04 2023

Keywords

Crossrefs

Cf. A366676, A367057, A367058, A367060, A367061, A367062.

Programs

  • PARI
    a(n) = sum(k=0, n\3, binomial(2*n-3*k+1, k)*binomial(3*n-6*k, n-3*k)/(2*n-3*k+1));

Formula

a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-3*k+1,k) * binomial(3*n-6*k,n-3*k)/(2*n-3*k+1).

A367060 G.f. satisfies A(x) = 1 + x*A(x)^3 + x^3*A(x)^4.

Original entry on oeis.org

1, 1, 3, 13, 62, 318, 1718, 9627, 55437, 326070, 1950630, 11831706, 72597453, 449804148, 2810260317, 17685019893, 111997074910, 713223954540, 4564502770117, 29341499243806, 189364923816282, 1226535071582818, 7970416067268898, 51949175133236526
Offset: 0

Views

Author

Seiichi Manyama, Nov 04 2023

Keywords

Crossrefs

Cf. A366676, A367057, A367058, A367059, A367061, A367062.

Programs

  • PARI
    a(n) = sum(k=0, n\3, binomial(2*n-2*k+1, k)*binomial(3*n-5*k, n-3*k)/(2*n-2*k+1));

Formula

a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-2*k+1,k) * binomial(3*n-5*k,n-3*k)/(2*n-2*k+1).

A367061 G.f. satisfies A(x) = 1 + x*A(x)^3 + x^3*A(x)^5.

Original entry on oeis.org

1, 1, 3, 13, 63, 328, 1797, 10210, 59607, 355409, 2155166, 13250055, 82402013, 517453773, 3276534510, 20897024350, 134118458191, 865574280977, 5613879001983, 36571135386965, 239187418784442, 1569994174618799, 10338925554033967, 68288387553861826
Offset: 0

Views

Author

Seiichi Manyama, Nov 04 2023

Keywords

Crossrefs

Cf. A366676, A367057, A367058, A367059, A367060, A367062.

Programs

  • PARI
    a(n) = sum(k=0, n\3, binomial(2*n-k+1, k)*binomial(3*n-4*k, n-3*k)/(2*n-k+1));

Formula

a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-k+1,k) * binomial(3*n-4*k,n-3*k)/(2*n-k+1).

A367062 G.f. satisfies A(x) = 1 + x*A(x)^3 + x^3*A(x)^6.

Original entry on oeis.org

1, 1, 3, 13, 64, 339, 1889, 10917, 64836, 393292, 2426335, 15176847, 96029114, 613540477, 3952727925, 25649572693, 167494312692, 1099850119488, 7257905610260, 48106858236044, 320131295055690, 2138010763838375, 14325505944147495, 96273042489762471
Offset: 0

Views

Author

Seiichi Manyama, Nov 04 2023

Keywords

Crossrefs

Cf. A366676, A367057, A367058, A367059, A367060, A367061.

Programs

  • PARI
    a(n) = sum(k=0, n\3, binomial(2*n+1, k)*binomial(3*n-3*k, n-3*k))/(2*n+1);

Formula

a(n) = (1/(2*n+1)) * Sum_{k=0..floor(n/3)} binomial(2*n+1,k) * binomial(3*n-3*k,n-3*k).

A367040 G.f. satisfies A(x) = 1 + x^2 + x*A(x)^3.

Original entry on oeis.org

1, 1, 4, 15, 70, 360, 1953, 11008, 63837, 378390, 2282205, 13960890, 86411232, 540166219, 3405341160, 21625820793, 138216775785, 888371346825, 5738510504979, 37234351046835, 242567430368298, 1585979835198675, 10403866383915844, 68453912880893025
Offset: 0

Views

Author

Seiichi Manyama, Nov 03 2023

Keywords

Crossrefs

Cf. A176697, A367041.
Cf. A366266, A366676.
Cf. A002293, A104979, A186997, A255673, A361245, A364474, A364475, A364478.

Programs

  • PARI
    a(n) = sum(k=0, n\2, binomial(2*(n-2*k)+1, k)*binomial(3*(n-2*k), n-2*k)/(2*(n-2*k)+1));

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(2*(n-2*k)+1,k) * binomial(3*(n-2*k),n-2*k)/(2*(n-2*k)+1).

A367042 G.f. satisfies A(x) = 1 + x^3 + x*A(x)^2.

Original entry on oeis.org

1, 1, 2, 6, 16, 48, 152, 500, 1688, 5816, 20368, 72288, 259424, 939808, 3432192, 12622416, 46706144, 173762016, 649569216, 2438748864, 9191656192, 34765298944, 131912452864, 501987944832, 1915417307392, 7326620001536, 28088736525824, 107913607531520
Offset: 0

Views

Author

Seiichi Manyama, Nov 03 2023

Keywords

Crossrefs

Cf. A366676, A367043.
Cf. A025227, A176697.
Cf. A226022.

Programs

  • PARI
    a(n) = sum(k=0, n\3, binomial(n-3*k+1, k)*binomial(2*(n-3*k), n-3*k)/(n-3*k+1));

Formula

G.f.: A(x) = 2*(1+x^3) / (1+sqrt(1-4*x*(1+x^3))).
a(n) = Sum_{k=0..floor(n/3)} binomial(n-3*k+1,k) * binomial(2*(n-3*k),n-3*k)/(n-3*k+1).

A367043 G.f. satisfies A(x) = 1 + x^3 + x*A(x)^4.

Original entry on oeis.org

1, 1, 4, 23, 144, 997, 7304, 55646, 436320, 3497846, 28538852, 236203518, 1978290648, 16735471979, 142789868112, 1227339581084, 10617748941840, 92377468226466, 807769888050640, 7095187345173620, 62574408414192220, 553881698543850337
Offset: 0

Views

Author

Seiichi Manyama, Nov 03 2023

Keywords

Crossrefs

Cf. A366676, A367042.

Programs

  • PARI
    a(n) = sum(k=0, n\3, binomial(3*(n-3*k)+1, k)*binomial(4*(n-3*k), n-3*k)/(3*(n-3*k)+1));

Formula

a(n) = Sum_{k=0..floor(n/3)} binomial(3*(n-3*k)+1,k) * binomial(4*(n-3*k),n-3*k)/(3*(n-3*k)+1).

A366677 G.f. satisfies A(x) = 1 + x^4 + x*A(x)^4.

Original entry on oeis.org

1, 1, 4, 22, 141, 973, 7112, 54040, 422552, 3377770, 27478568, 226753828, 1893462584, 15969598554, 135842638632, 1164075017512, 10039732285528, 87081507756245, 759128176746864, 6647475055207618, 58445784269830824, 515745587816906733
Offset: 0

Views

Author

Seiichi Manyama, Oct 16 2023

Keywords

Crossrefs

Cf. A176697, A366676.
Cf. A366267, A366558.

Programs

  • PARI
    a(n) = sum(k=0, n\4, binomial(3*(n-4*k)+1, k)*binomial(4*(n-4*k), n-4*k)/(3*(n-4*k)+1));

Formula

a(n) = Sum_{k=0..floor(n/4)} binomial(3*(n-4*k)+1,k) * binomial(4*(n-4*k),n-4*k)/(3*(n-4*k)+1).
Showing 1-10 of 11 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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