A352123 - OEIS

%I #13 Mar 06 2022 08:41:21

%S 1,1,-7,73,-1135,24241,-659767,21796153,-846456415,37772943841,

%T -1904103268327,106992035096233,-6630198107231695,449171668238551441,

%U -33024202381308836887,2618743082761141212313,-222782402553043700662975,20238957866498067052271041

%N Expansion of e.g.f. (2 - exp(-4*x))^(1/4).

%F a(n) = Sum_{k=0..n} (-4)^(n-k) * (Product_{j=0..k-1} (-4*j+1)) * Stirling2(n,k).

%F a(n) ~ n! * (-1)^(n+1) * Gamma(1/4) * 2^(2*n - 9/4) / (Pi * n^(5/4) * log(2)^(n -1/4)). - _Vaclav Kotesovec_, Mar 06 2022

%t m = 17; Range[0, m]! * CoefficientList[Series[(2 - Exp[-4*x])^(1/4), {x, 0, m}], x] (* _Amiram Eldar_, Mar 05 2022 *)

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace((2-exp(-4*x))^(1/4)))

%o (PARI) a(n) = sum(k=0, n, (-4)^(n-k)*prod(j=0, k-1, -4*j+1)*stirling(n, k, 2));

%Y Cf. A352121, A352122.

%Y Cf. A352114.

%K sign

%O 0,3

%A _Seiichi Manyama_, Mar 05 2022