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 1-5 of 5 results.

A364522 G.f. satisfies A(x) = 1 + x*A(x) + x^5*A(x)^5.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 7, 22, 57, 127, 258, 518, 1123, 2718, 7008, 18054, 44969, 108189, 255919, 609179, 1482210, 3689155, 9294440, 23419705, 58639835, 145948111, 362721386, 904673836, 2270287636, 5729191861, 14502873988, 36735974548, 93001413353, 235372519273
Offset: 0

Views

Author

Seiichi Manyama, Jul 27 2023

Keywords

Comments

Number of ordered trees with n edges and having nonleaf nodes of outdegrees 1 or 5. - Emanuele Munarini, Jul 11 2024

Links

Crossrefs

Cf. A001006, A071879, A127902, A364523.
Cf. A182454, A349311.

Programs

  • PARI
    a(n) = sum(k=0, n\5, binomial(n, 5*k)*binomial(5*k, k)/(4*k+1));

Formula

a(n) = Sum_{k=0..floor(n/5)} binomial(n,5*k) * binomial(5*k,k) / (4*k+1).

A365760 G.f. satisfies A(x) = 1 + x*A(x)*(1 + x^4*A(x)^5).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 8, 29, 85, 211, 469, 1003, 2263, 5734, 15926, 45188, 124730, 330583, 850783, 2175406, 5650746, 15064128, 41006034, 112492472, 307511726, 833907512, 2247908392, 6056190352, 16390505332, 44659671982, 122380777306, 336326321179, 924529751087
Offset: 0

Views

Author

Seiichi Manyama, Sep 18 2023

Keywords

Crossrefs

Cf. A364472, A364523, A365759, A365761.
Cf. A365758.

Programs

  • PARI
    a(n) = sum(k=0, n\5, binomial(n-4*k, k)*binomial(n+k+1, n-4*k)/(n+k+1));

Formula

a(n) = Sum_{k=0..floor(n/5)} binomial(n-4*k,k) * binomial(n+k+1,n-4*k) / (n+k+1) = Sum_{k=0..floor(n/5)} binomial(n+k,6*k) * binomial(6*k,k) / (5*k+1).

A365759 G.f. satisfies A(x) = 1 + x*A(x)*(1 + x^3*A(x)^5).

Original entry on oeis.org

1, 1, 1, 1, 2, 8, 29, 85, 217, 541, 1471, 4447, 13975, 43103, 129083, 382535, 1147956, 3519462, 10947483, 34162483, 106341406, 330590764, 1030528133, 3229411337, 10170424724, 32127163822, 101633409379, 321862281571, 1020889305476, 3244779281894, 10335256815761
Offset: 0

Views

Author

Seiichi Manyama, Sep 18 2023

Keywords

Crossrefs

Cf. A364472, A364523, A365760, A365761.
Cf. A365757.

Programs

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

Formula

a(n) = Sum_{k=0..floor(n/4)} binomial(n-3*k,k) * binomial(n+2*k+1,n-3*k) / (n+2*k+1) = Sum_{k=0..floor(n/4)} binomial(n+2*k,6*k) * binomial(6*k,k) / (5*k+1).

A365761 G.f. satisfies A(x) = 1 + x*A(x)*(1 + x^2*A(x)^5).

Original entry on oeis.org

1, 1, 1, 2, 8, 29, 91, 289, 1009, 3706, 13606, 49822, 184726, 696052, 2648746, 10132072, 38952970, 150635860, 585724594, 2287631614, 8968247626, 35281363830, 139256375922, 551306272137, 2188516471579, 8709331962133, 34739262293455, 138863195368540
Offset: 0

Views

Author

Seiichi Manyama, Sep 18 2023

Keywords

Crossrefs

Cf. A364472, A364523, A365759, A365760.
Cf. A365693.

Programs

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

Formula

a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k,k) * binomial(n+3*k+1,n-2*k) / (n+3*k+1) = Sum_{k=0..floor(n/3)} binomial(n+3*k,6*k) * binomial(6*k,k) / (5*k+1).

A365798 G.f. satisfies A(x) = 1 + x*A(x)*(1 + x^5*A(x)^4).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 7, 22, 57, 127, 253, 468, 848, 1618, 3433, 8009, 19384, 46264, 106369, 235179, 505955, 1079790, 2332555, 5166405, 11737860, 27086236, 62676956, 144074416, 327837356, 739787486, 1663922487, 3751649542, 8513640107, 19464624667
Offset: 0

Views

Author

Seiichi Manyama, Sep 19 2023

Keywords

Crossrefs

Cf. A063019, A182454, A349311, A364539, A364522.
Cf. A005708, A364523, A365732, A365733.
Cf. A002294, A365736.

Programs

  • PARI
    a(n) = sum(k=0, n\6, binomial(n-5*k, k)*binomial(n-k+1, n-5*k)/(n-k+1));

Formula

a(n) = Sum_{k=0..floor(n/6)} binomial(n-5*k,k) * binomial(n-k+1,n-5*k) / (n-k+1) = Sum_{k=0..floor(n/6)} binomial(n-k,5*k) * binomial(5*k,k) / (4*k+1).
Showing 1-5 of 5 results.

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

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