A354015 Expansion of e.g.f. 1/(1 - x)^(1 - log(1-x)).

1, 1, 4, 18, 106, 750, 6188, 58184, 613156, 7149780, 91319712, 1267089912, 18969355656, 304646227704, 5222700792528, 95169251327040, 1836450816902928, 37403582826055824, 801728489886598848, 18037821249349491360, 424970923585819603872, 10462258547232790348512

Offset: 0

Author

Seiichi Manyama, May 14 2022

Keywords

• nonn

Crossrefs

Cf. A000776, A047974, A189423, A353995, A354013.

Programs

PARI

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

PARI

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

PARI

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

PARI

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

Formula

E.g.f.: exp( -log(1-x) * (1 - log(1-x)) ).

a(0) = 1; a(n) = Sum_{k=1..n} A000776(k-1) * binomial(n-1,k-1) * a(n-k) = (n-1)! * Sum_{k=1..n} (1 + 2*Sum_{j=1..k-1} 1/j) * a(n-k)/(n-k)!.

a(n) = Sum_{k=0..n} A047974(k) * |Stirling1(n,k)|.