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

%I #7 Dec 04 2023 06:35:55

%S 1,0,6,9,228,1095,23238,215481,4657992,66216555,1553967210,

%T 29793656013,777115661292,18608934688383,542832959656302,

%U 15470567460571905,503794462155308688,16557037363336856019,598704921471691072242,22205328374455141122165

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

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

%F a(n) = n! * Sum_{k=0..floor(n/2)} 3^k * k! * Stirling2(n-k,k)/(n-k)!.

%o (PARI) a(n) = n!*sum(k=0, n\2, 3^k*k!*stirling(n-k, k, 2)/(n-k)!);

%Y Cf. A052848, A367880.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Dec 03 2023