%I #14 May 30 2024 11:07:57

%S 1,3,27,369,6849,160803,4566987,152204769,5822610849,251445000483,

%T 12098060349147,641736701136369,37204969609266849,2340437711290748163,

%U 158770522442243864907,11553653430580844747169,897732793887437892390849,74182365989862425679675843

%N Expansion of e.g.f. 1 / (3 - 2 * exp(2*x))^(3/4).

%F a(n) = Sum_{k=0..n} 2^(n-k) * (Product_{j=0..k-1} (4*j+3)) * Stirling2(n,k).

%F a(0) = 1; a(n) = Sum_{k=1..n} 2^k * (2 - 1/2 * k/n) * binomial(n,k) * a(n-k).

%F a(0) = 1; a(n) = 3*a(n-1) - 3*Sum_{k=1..n-1} (-2)^k * binomial(n-1,k) * a(n-k).

%t With[{nn=20},CoefficientList[Series[1/(3-2Exp[2x])^(3/4),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, May 30 2024 *)

%o (PARI) a(n) = sum(k=0, n, 2^(n-k)*prod(j=0, k-1, 4*j+3)*stirling(n, k, 2));

%Y Cf. A365777, A365782.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Nov 16 2023