A354504 Expansion of e.g.f. ( Product_{k>0} (1 + x^k)^k )^exp(x).

1, 1, 6, 48, 402, 4375, 54595, 777189, 12284188, 215999025, 4132338673, 85640640877, 1910121348674, 45571124446445, 1157169377895739, 31150000798832647, 885481496002286200, 26498034473000080321, 832407848080194500301, 27378188500890922864153

Offset: 0

Author

Seiichi Manyama, Aug 15 2022

Keywords

• nonn

Crossrefs

Cf. A347915, A354503.

Cf. A354508, A356394.

Programs

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1+x^k)^k)^exp(x)))

PARI

a354508(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*d^2)/(k*(n-k)!));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a354508(j)*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A354508(k) * binomial(n-1,k-1) * a(n-k).