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.

A361532 Expansion of e.g.f. exp((x + x^2/2)/(1-x)).

Original entry on oeis.org

1, 1, 4, 19, 118, 886, 7786, 78184, 881644, 11017108, 150966856, 2249261356, 36181351504, 624658612384, 11516406883528, 225740649754936, 4686671645814736, 102712289940757264, 2369128149877075264, 57359541280704038128, 1454229915957292684576
Offset: 0

Views

Author

Seiichi Manyama, Mar 15 2023

Keywords

Crossrefs

Cf. A361548, A361557, A361558.
Cf. A052845, A112242, A185369.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[Exp[(x+x^2/2)/(1-x)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 08 2023 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((x+x^2/2)/(1-x))))

Formula

a(n) = (2*n-1) * a(n-1) - (n-1)*(n-3) * a(n-2) - binomial(n-1,2) * a(n-3) for n > 2.
a(n) ~ 2^(-3/4) * 3^(1/4) * exp(-5/4 + sqrt(6*n) - n) * n^(n - 1/4) * (1 + sqrt(3)/(2*sqrt(2*n))). - Vaclav Kotesovec, Mar 20 2023

A361548 Expansion of e.g.f. exp((x + x^2/2 + x^3/6)/(1-x)).

Original entry on oeis.org

1, 1, 4, 20, 126, 966, 8656, 88544, 1016380, 12920156, 179996816, 2725070096, 44521522024, 780344770440, 14599772973696, 290311643773376, 6112190642062096, 135798496839920144, 3174483084427144000, 77872118431269176896, 1999809157085214044896
Offset: 0

Views

Author

Seiichi Manyama, Mar 15 2023

Keywords

Crossrefs

Cf. A361532, A361557, A361558.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[Exp[(x+x^2/2+x^3/6)/(1-x)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Nov 03 2024 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((x+x^2/2+x^3/6)/(1-x))))

Formula

a(n) = (2*n-1) * a(n-1) - (n-1)*(n-3) * a(n-2) - 2*binomial(n-1,3) * a(n-4) for n > 3.

A361557 Expansion of e.g.f. exp((exp(x) - 1)/(1-x)).

Original entry on oeis.org

1, 1, 4, 20, 127, 977, 8789, 90267, 1040260, 13275258, 185653535, 2821321725, 46265262553, 813871304989, 15281792484768, 304949014412540, 6442741397501699, 143633948442619765, 3369004776395733829, 82919378806522132407, 2136425765494805888952
Offset: 0

Views

Author

Seiichi Manyama, Mar 15 2023

Keywords

Crossrefs

Cf. A361532, A361548, A361558.
Cf. A002627.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((exp(x)-1)/(1-x))))

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A002627(k) * binomial(n-1,k-1) * a(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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