A355718 Expansion of e.g.f. exp( x/(1 + log(1-x)) ).

1, 1, 3, 16, 117, 1071, 11725, 149122, 2158401, 35006941, 628552231, 12372376116, 264849067549, 6124239060915, 152099146415385, 4037206919213686, 114038575520545153, 3415098936831144537, 108065651366801837611, 3602585901321224507992

Offset: 0

Author

Seiichi Manyama, Jul 15 2022

Keywords

• nonn

Crossrefs

Cf. A355719, A355720.

Cf. A007840, A052860.

Programs

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x/(1+log(1-x)))))

PARI

a007840(n) = sum(k=0, n, k!*abs(stirling(n, k, 1)));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, j*a007840(j-1)*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A052860(k) * binomial(n-1,k-1) * a(n-k).

a(n) ~ n^(n-1/4) * exp(1/4 - exp(-1) + 2*exp(-1/2)*sqrt(n)) / (sqrt(2) * (exp(1) - 1)^n). - Vaclav Kotesovec, Jul 15 2022