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 1-2 of 2 results.

A354325 Expansion of e.g.f. 1/(1 - x/4 * (exp(2 * x) - 1)).

Original entry on oeis.org

1, 0, 1, 3, 14, 80, 558, 4522, 41864, 436032, 5046680, 64251176, 892361520, 13426491520, 217555171568, 3776935252560, 69942048682112, 1376150998836224, 28669321699355520, 630448829825395840, 14593473117397510400, 354696400190943197184, 9031466708133617225984
Offset: 0

Views

Author

Seiichi Manyama, May 24 2022

Keywords

Crossrefs

Cf. A354313, A354323, A354326.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[1/(1-x/4 (Exp[2x]-1)),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Dec 02 2022 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x/4*(exp(2*x)-1))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j*2^(j-3)*binomial(i, j)*v[i-j+1])); v;
  • PARI
    a(n) = n!*sum(k=0, n\2, 2^(n-3*k)*k!*stirling(n-k, k, 2)/(n-k)!);

Formula

a(0) = 1; a(n) = Sum_{k=2..n} k * 2^(k-3) * binomial(n,k) * a(n-k).
a(n) = n! * Sum_{k=0..floor(n/2)} 2^(n-3*k) * k! * Stirling2(n-k,k)/(n-k)!.

A354319 Expansion of e.g.f. 1/(1 - 2*x)^(x/4).

Original entry on oeis.org

1, 0, 1, 3, 19, 150, 1497, 17955, 251681, 4036284, 72874125, 1462571055, 32297755803, 778188449610, 20313917363733, 571081958323695, 17201321168216385, 552635193533958360, 18863471310967732473, 681711909339186154395, 26003437607893415476995
Offset: 0

Views

Author

Seiichi Manyama, May 24 2022

Keywords

Crossrefs

Cf. A354309, A354323, A354327.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-2*x)^(x/4)))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=2, i, j*2^(j-3)/(j-1)*v[i-j+1]/(i-j)!)); v;
  • PARI
    a(n) = n!*sum(k=0, n\2, 2^(n-3*k)*abs(stirling(n-k, k, 1))/(n-k)!);

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=2..n} k * 2^(k-3)/(k-1) * a(n-k)/(n-k)!.
a(n) = n! * Sum_{k=0..floor(n/2)} 2^(n-3*k) * |Stirling1(n-k,k)|/(n-k)!.
a(n) ~ sqrt(Pi) * 2^(n + 1/2) * n^(n - 3/8) / (Gamma(1/8) * exp(n)). - Vaclav Kotesovec, Mar 14 2024
Showing 1-2 of 2 results.

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