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 A354241 Expansion of e.g.f. 1/sqrt(1 + 4 * log(1-x)). 7

%I #19 Sep 10 2023 08:39:29

%S 1,2,14,160,2544,51888,1292208,38012448,1289847456,49593778368,

%T 2130914229312,101188640375040,5262325852773120,297450338175682560,

%U 18157597034693207040,1190483599149657584640,83433723762978141189120,6224485980052510972692480

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

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

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

%F a(n) ~ n^n / (sqrt(2) * (exp(1/4)-1)^(n + 1/2) * exp(3*n/4)). - _Vaclav Kotesovec_, Jun 04 2022

%F 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

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

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

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

%Y Cf. A354240, A354242, A354244.

%K nonn

%O 0,2

%A _Seiichi Manyama_, May 20 2022

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Last modified December 3 05:03 EST 2023. Contains 367531 sequences. (Running on oeis4.)