A354260 Expansion of e.g.f. 1/sqrt(1 - 8 * log(1+x)).

1, 4, 44, 824, 21624, 730176, 30144192, 1470979968, 82833047424, 5286741547008, 377135779749888, 29736359948175360, 2568013599548037120, 241061197802997288960, 24439230397588083240960, 2661258811775918180474880, 309780832909692738794987520

Offset: 0

Author

Seiichi Manyama, May 21 2022

Keywords

• nonn

Crossrefs

Cf. A320343, A354240, A354259.

Cf. A354253, A354262.

Programs

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/sqrt(1-8*log(1+x))))

PARI

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

PARI

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

Formula

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

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

a(n) ~ n^n / (2 * (exp(1/8)-1)^(n + 1/2) * exp(n - 1/16)). - Vaclav Kotesovec, Jun 04 2022