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 1-6 of 6 results.

A364523 G.f. satisfies A(x) = 1 + x*A(x) + x^6*A(x)^6.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 8, 29, 85, 211, 463, 931, 1795, 3550, 7736, 18929, 49505, 130000, 330430, 804271, 1885675, 4327555, 9929515, 23224435, 55907251, 138016906, 345107296, 862546231, 2136402451, 5231163232, 12697101118, 30723857209, 74569942745
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 6. - Emanuele Munarini, Jul 11 2024

Links

Crossrefs

Cf. A001006, A071879, A127902, A364522.
Cf. A349312, A364472.

Programs

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

Formula

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

A369128 Expansion of (1/x) * Series_Reversion( x / ((1+x)^5+x^5) ).

Original entry on oeis.org

1, 5, 35, 285, 2530, 23752, 231910, 2331040, 23960235, 250692365, 2661086895, 28587333725, 310217791590, 3395464391870, 37442295427120, 415570885425280, 4638842010800025, 52044582325415025, 586553425250933055, 6637525235622842585, 75387741117556006435
Offset: 0

Views

Author

Seiichi Manyama, Jan 14 2024

Keywords

Links

Crossrefs

Cf. A071356, A192132, A369126.
Cf. A364522.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)^5+x^5))/x)
  • PARI
    a(n) = sum(k=0, n\5, binomial(n+1, k)*binomial(5*n-5*k+5, n-5*k))/(n+1);

Formula

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

A364539 G.f. satisfies A(x) = 1 + x*A(x) + x^3*A(x)^5.

Original entry on oeis.org

1, 1, 1, 2, 7, 22, 62, 182, 583, 1928, 6358, 21063, 70888, 241889, 831634, 2874584, 9995579, 34966279, 122938956, 434062141, 1538378816, 5471697241, 19525345791, 69880082323, 250767909528, 902123110483, 3252793321513, 11753570922933, 42553831219830
Offset: 0

Views

Author

Seiichi Manyama, Jul 28 2023

Keywords

Crossrefs

Cf. A063019, A182454, A349311, A364522.
Cf. A023431, A071879, A215340.

Programs

  • PARI
    a(n) = sum(k=0, n\3, binomial(n+2*k, 5*k)*binomial(5*k, k)/(4*k+1));

Formula

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

A366057 Expansion of (1/x) * Series_Reversion( x/(1-x+x^5) ).

Original entry on oeis.org

1, -1, 1, -1, 1, 0, -5, 20, -55, 125, -246, 406, -461, -144, 3004, -11978, 35113, -86293, 181663, -314603, 365922, 150023, -2696308, 10969573, -32970453, 82976409, -178372934, 314133884, -367436684, -179661091, 2923282216, -11972239216, 36369188841, -92517132841
Offset: 0

Views

Author

Seiichi Manyama, Sep 27 2023

Keywords

Crossrefs

Cf. A364522, A365702, A366051.

Programs

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

Formula

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

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).

A369688 G.f. satisfies A(x) = 1 + x*A(x) + x^2*(1-x)^3*A(x)^5.

Original entry on oeis.org

1, 1, 2, 4, 12, 36, 126, 442, 1644, 6172, 23801, 92731, 366688, 1462852, 5891808, 23898576, 97600556, 400844140, 1654818768, 6862550360, 28576414261, 119434041561, 500849380048, 2106740001442, 8886482895068, 37580609774876, 159303913630686
Offset: 0

Views

Author

Seiichi Manyama, Jan 28 2024

Keywords

Crossrefs

Cf. A227035, A346647, A364522, A364597.
Cf. A049130, A364594.

Programs

  • PARI
    a(n) = sum(k=0, n\2, binomial(n, 2*k)*binomial(5*k, k)/(4*k+1));

Formula

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

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