A353119 Expansion of e.g.f. 1/(1 - log(1 - x)^4).

1, 0, 0, 0, 24, 240, 2040, 17640, 202776, 3066336, 52446720, 933636000, 17416490784, 350580364992, 7719355635264, 184232862777600, 4691944607751936, 126358891201529856, 3591751011211717632, 107772466927523060736, 3408777017097439186944

Offset: 0

Author

Seiichi Manyama, May 06 2022

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..418

Crossrefs

Cf. A052811, A353118, A353200.

Cf. A346923.

Programs

Mathematica

With[{nn=20},CoefficientList[Series[1/(1-Log[1-x]^4),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 13 2024 *)

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-log(1-x)^4)))

PARI

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=24*sum(j=1, i, binomial(i, j)*abs(stirling(j, 4, 1))*v[i-j+1])); v;

PARI

a(n) = sum(k=0, n\4, (4*k)!*abs(stirling(n, 4*k, 1)));

Formula

a(0) = 1; a(n) = 24 * Sum_{k=1..n} binomial(n,k) * |Stirling1(k,4)| * a(n-k).

a(n) = Sum_{k=0..floor(n/4)} (4*k)! * |Stirling1(n,4*k)|.

a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / (4 * (exp(1) - 1)^(n+1)). - Vaclav Kotesovec, May 07 2022