A354262 Expansion of e.g.f. 1/sqrt(1 + 8 * log(1-x)).

1, 4, 52, 1112, 33192, 1272576, 59607552, 3298935552, 210638509824, 15241340093952, 1232504690492928, 110154484622208000, 10782300230031713280, 1147157496053856645120, 131810751499551281786880, 16266976762439018716323840, 2145960434809665656603320320

Offset: 0

Author

Seiichi Manyama, May 21 2022

Keywords

• nonn

Crossrefs

Cf. A346978, A354241, A354261.

Cf. A354253, A354260.

Programs

Mathematica

With[{nn=20},CoefficientList[Series[1/Sqrt[1+8*Log[1-x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Dec 14 2024 *)

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)!*abs(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(7*n/8)). - Vaclav Kotesovec, Jun 04 2022