A354241 Expansion of e.g.f. 1/sqrt(1 + 4 * log(1-x)).

1, 2, 14, 160, 2544, 51888, 1292208, 38012448, 1289847456, 49593778368, 2130914229312, 101188640375040, 5262325852773120, 297450338175682560, 18157597034693207040, 1190483599149657584640, 83433723762978141189120, 6224485980052510972692480

Offset: 0

Author

Seiichi Manyama, May 20 2022

Keywords

• nonn

Crossrefs

Cf. A354240, A354242, A354244.

Programs

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/sqrt(1+4*log(1-x))))

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, binomial(2*k, k)*(-log(1-x))^k)))

PARI

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

Formula

E.g.f.: Sum_{k>=0} binomial(2*k,k) * (-log(1-x))^k.

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

a(n) ~ n^n / (sqrt(2) * (exp(1/4)-1)^(n + 1/2) * exp(3*n/4)). - Vaclav Kotesovec, Jun 04 2022

a(0) = 1; a(n) = Sum_{k=1..n} (4 - 2*k/n) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 09 2023