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 1-3 of 3 results.

A365843 Expansion of (1/x) * Series_Reversion( x*(1-x)^3/(1+x)^3 ).

Original entry on oeis.org

1, 6, 54, 578, 6810, 85278, 1113854, 15004746, 206955378, 2908113974, 41484917958, 599202514578, 8745727050762, 128790559374030, 1911191826600462, 28551332345784730, 429040549473424866, 6480799118506040934, 98349636147075506006, 1498732955394826784226
Offset: 0

Views

Author

Seiichi Manyama, Sep 20 2023

Keywords

Crossrefs

Column k=3 of A378238.
Cf. A007297, A365842, A365844, A365845.
Cf. A032349, A365622, A365847.
Cf. A144097.

Programs

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

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(3*n+k+2,k) * binomial(3*(n+1),n-k).
G.f.: B^3, where B is the g.f. of A144097.
a(n) ~ sqrt(8060 + 2651*sqrt(10)) * (223 + 70*sqrt(10))^n / (2 * sqrt(5*Pi) * n^(3/2) * 3^(3*n + 5/2)). - Vaclav Kotesovec, Nov 28 2024

A365842 Expansion of (1/x) * Series_Reversion( x*(1-x)^2/(1+x)^3 ).

Original entry on oeis.org

1, 5, 37, 325, 3141, 32261, 345605, 3818501, 43197445, 497868805, 5825331205, 69013667845, 826213203973, 9979713945605, 121472752156677, 1488482728148997, 18346810389299205, 227319830355640325, 2829629321065267205, 35369618935665131525, 443775430273133445125
Offset: 0

Views

Author

Seiichi Manyama, Sep 20 2023

Keywords

Crossrefs

Cf. A007297, A365843, A365844, A365845.

Programs

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

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(2*n+k+1,k) * binomial(3*(n+1),n-k).
a(n) = (1/(n+1)) * [x^n] ( (1+x)^3 / (1-x)^2 )^(n+1). - Seiichi Manyama, Feb 17 2024

A365845 Expansion of (1/x) * Series_Reversion( x*(1-x)^5/(1+x)^3 ).

Original entry on oeis.org

1, 8, 97, 1400, 22243, 375584, 6614508, 120136984, 2234022775, 42322629960, 813939319697, 15849232257824, 311858145053076, 6191083938051840, 123852349440862504, 2494251111318893400, 50526944132627936127, 1028872756710478785560
Offset: 0

Views

Author

Seiichi Manyama, Sep 20 2023

Keywords

Crossrefs

Cf. A007297, A365842, A365843, A365844.

Programs

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

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(5*n+k+4,k) * binomial(3*(n+1),n-k).
Showing 1-3 of 3 results.

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

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