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.

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A378461 a(n) = Sum_{k=0..n} binomial(n+k-1,k) * binomial(2*n+2*k-1,n-k).

Original entry on oeis.org

1, 2, 16, 137, 1216, 11057, 102229, 956601, 9032680, 85893860, 821402341, 7891371303, 76105710253, 736364519399, 7144586617597, 69487754788517, 677259385478616, 6613163312601491, 64681617534027028, 633569272646345064, 6214190349161222941, 61023489213944162889
Offset: 0

Views

Author

Seiichi Manyama, Nov 27 2024

Keywords

Crossrefs

Cf. A009723, A378460.
Cf. A378466.

Programs

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

Formula

a(n) = [x^n] 1/(1 - x - x/(1 - x)^2)^n.

A378465 Expansion of (1/x) * Series_Reversion( x * (1 - x - x/(1 - x)) ).

Original entry on oeis.org

1, 2, 9, 51, 324, 2206, 15737, 116098, 878495, 6780544, 53175176, 422508607, 3394004192, 27518168434, 224899980185, 1850830170355, 15324273361220, 127562500961502, 1066940307951747, 8962213871074848, 75572666059970392, 639485384767169924, 5428457500063304272
Offset: 0

Views

Author

Seiichi Manyama, Nov 27 2024

Keywords

Links

Crossrefs

Cf. A151374, A378466.
Cf. A378460.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x-x/(1-x)))/x)
  • PARI
    a(n) = sum(k=0, n, binomial(n+k, k)*binomial(2*n+k, n-k))/(n+1);

Formula

G.f.: exp( Sum_{k>=1} A378460(k) * x^k/k ).
a(n) = (1/(n+1)) * [x^n] 1/(1 - x - x/(1 - x))^(n+1).
a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+k,k) * binomial(2*n+k,n-k).
a(n) ~ ((16 + 12*2^(1/3) + 9*2^(2/3))/5)^n / (sqrt(6*(4 - 3*2^(1/3))*Pi) * n^(3/2)). - Vaclav Kotesovec, Nov 27 2024
Showing 1-2 of 2 results.

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

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