A353118 Expansion of e.g.f. 1/(1 + log(1 - x)^3).

1, 0, 0, 6, 36, 210, 2070, 24864, 310632, 4337544, 68922360, 1205002656, 22844264256, 469287123552, 10397824478496, 246800350393344, 6246190572981120, 167972669001740160, 4783274802508890240, 143775432034543203840, 4548946867429143444480

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, A353119, A353200.

Cf. A346922.

Programs

Mathematica

With[{nn=20},CoefficientList[Series[1/(1+Log[1-x]^3),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 04 2023 *)

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+log(1-x)^3)))

PARI

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=6*sum(j=1, i, binomial(i, j)*abs(stirling(j, 3, 1))*v[i-j+1])); v;

PARI

a(n) = sum(k=0, n\3, (3*k)!*abs(stirling(n, 3*k, 1)));

Formula

a(0) = 1; a(n) = 6 * Sum_{k=1..n} binomial(n,k) * |Stirling1(k,3)| * a(n-k).

a(n) = Sum_{k=0..floor(n/3)} (3*k)! * |Stirling1(n,3*k)|.

a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / (3 * (exp(1) - 1)^(n+1)). - Vaclav Kotesovec, May 07 2022