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

%I #9 Aug 25 2024 09:58:19

%S 1,0,0,0,8,20,80,420,11648,100800,912000,9055200,181547520,2790627840,

%T 41568334080,635617382400,13172198645760,273158953267200,

%U 5632405756723200,117530452124467200,2815021136030515200,71252240659839590400,1844362570865444044800

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

%F a(n) = n! * Sum_{k=0..floor(n/4)} (Product_{j=0..k-1} (6*j+2)) * |Stirling1(n-3*k,k)|/(6^k*(n-3k)!).

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

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

%Y Cf. A351504, A375699, A375701.

%K nonn

%O 0,5

%A _Seiichi Manyama_, Aug 25 2024