A351503 Expansion of e.g.f. 1/(1 + x^2 * log(1 - x)).

1, 0, 0, 6, 12, 40, 900, 6048, 43680, 717120, 8658720, 102231360, 1735525440, 28819964160, 473955850368, 9235543363200, 189202617676800, 3940225003653120, 89804740509434880, 2169337606086389760, 54085753764912844800, 1429100881569205125120

Offset: 0

Author

Seiichi Manyama, May 04 2022

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..433

Crossrefs

Cf. A052830, A351504.

Cf. A351505, A353228.

Programs

Mathematica

With[{nn=30},CoefficientList[Series[1/(1+x^2 Log[1-x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 18 2024 *)

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+x^2*log(1-x))))

PARI

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i!*sum(j=3, i, 1/(j-2)*v[i-j+1]/(i-j)!)); v;

PARI

a(n) = n!*sum(k=0, n\3, k!*abs(stirling(n-2*k, k, 1))/(n-2*k)!);

Formula

a(0) = 1; a(n) = n! * Sum_{k=3..n} 1/(k-2) * a(n-k)/(n-k)!.

a(n) = n! * Sum_{k=0..floor(n/3)} k! * |Stirling1(n-2*k,k)|/(n-2*k)!.