A189423 Expansion of e.g.f. exp(log(1+x) + log(1+x)^2).

1, 1, 2, 0, 10, -50, 368, -3052, 28740, -302220, 3508152, -44532048, 613399752, -9109006920, 145029146208, -2463935369040, 44482964644368, -850291412311152, 17153458120885152, -364163960169826944, 8114899768747511712, -189364681355153357088, 4617713773733245962240

Offset: 0

Author

Vladimir Kruchinin, Apr 21 2011

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..447
• Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Crossrefs

Cf. A005444, A047974.

Programs

Maxima

a(n):=sum(sum(k!*binomial(m,k-m)*stirling1(n,k),k,m,n)/m!,m,1,n);

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(log(1+x)*(1+log(1+x))))) \\ Seiichi Manyama, May 14 2022

PARI

a(n) = sum(k=0, n, k!*sum(j=0, k\2, 1/(j!*(k-2*j)!))*stirling(n, k, 1)); \\ Seiichi Manyama, May 14 2022

Formula

a(0) = 1; a(n) = Sum_{m=1..n} Sum_{k=m..n} k!*binomial(m,k-m)*stirling1(n,k)/m! for n>0.

a(n) = Sum_{k=0..n} A047974(k) * Stirling1(n,k). - Seiichi Manyama, May 14 2022