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Expansion of e.g.f. 1 / (1 - x^3 * (exp(x) - 1))^(1/6).
2

%I #7 Aug 25 2024 09:58:30

%S 1,0,0,0,4,10,20,35,3976,35364,205920,970365,37643980,670990606,

%T 7705037704,69043474955,1690055888080,43135620048200,793592298255936,

%U 11401734214307769,250361353418216340,7380072768323315410,187670442928777057480,3868359812089009616071

%N Expansion of e.g.f. 1 / (1 - x^3 * (exp(x) - 1))^(1/6).

%F a(n) = n! * Sum_{k=0..floor(n/4)} (Product_{j=0..k-1} (6*j+1)) * Stirling2(n-3*k,k)/(6^k*(n-3k)!).

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x^3*(exp(x)-1))^(1/6)))

%o (PARI) a(n) = n!*sum(k=0, n\4, prod(j=0, k-1, 6*j+1)*stirling(n-3*k, k, 2)/(6^k*(n-3*k)!));

%Y Cf. A358014, A375717, A375718.

%K nonn

%O 0,5

%A _Seiichi Manyama_, Aug 25 2024