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A356813 Expansion of e.g.f. exp(x * (1 - exp(3*x))).

%I A356813 #20 Feb 23 2023 18:03:42
%S A356813 1,0,-6,-27,0,1215,12312,45927,-657072,-15857937,-167699160,
%T A356813 -266960529,29356170984,700068823623,8419188469104,-1491045413265,
%U A356813 -2856006296224992,-79065447339366945,-1162293393139510824,-744123842820101745,538503788896323210360
%N A356813 Expansion of e.g.f. exp(x * (1 - exp(3*x))).
%H A356813 Seiichi Manyama, <a href="/A356813/b356813.txt">Table of n, a(n) for n = 0..470</a>
%F A356813 G.f.: Sum_{k>=0} (-x)^k / (1 - (3*k+1)*x)^(k+1).
%F A356813 a(n) = Sum_{k=0..n} (-1)^k * (3*k+1)^(n-k) * binomial(n,k).
%F A356813 a(n) = n! * Sum_{k=0..floor(n/2)} (-1)^k * 3^(n-k) * Stirling2(n-k,k)/(n-k)!.
%o A356813 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(1-exp(3*x)))))
%o A356813 (PARI) my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (-x)^k/(1-(3*k+1)*x)^(k+1)))
%o A356813 (PARI) a(n) = sum(k=0, n, (-1)^k*(3*k+1)^(n-k)*binomial(n, k));
%o A356813 (PARI) a(n) = n!*sum(k=0, n\2, (-1)^k*3^(n-k)*stirling(n-k, k, 2)/(n-k)!);
%Y A356813 Cf. A292893, A356812.
%Y A356813 Cf. A351737, A356816.
%K A356813 sign
%O A356813 0,3
%A A356813 _Seiichi Manyama_, Aug 29 2022