A354240 Expansion of e.g.f. 1/sqrt(1 - 4 * log(1+x)).
1, 2, 10, 88, 1080, 17088, 330528, 7558752, 199487136, 5967529152, 199533657792, 7374470138880, 298520508249600, 13135454575464960, 624240306760343040, 31864146725023718400, 1738698154646011499520, 100996114388088994007040
Offset: 0
Keywords
Programs
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Mathematica
With[{nn=20},CoefficientList[Series[1/Sqrt[1-4Log[1+x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 04 2025 *) -
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)!*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(n - 1/8)). - Vaclav Kotesovec, Jun 04 2022
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * (4 - 2*k/n) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 11 2023