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A351737 Expansion of e.g.f. exp( x * (exp(3 * x) - 1) ).

%I A351737 #21 Aug 29 2022 16:35:46
%S A351737 1,0,6,27,216,2025,21708,260253,3460320,50395041,795324420,
%T A351737 13495904829,244747554912,4718754452529,96285948702804,
%U A351737 2071265238290565,46815054101658432,1108489016781839169,27424412680091114628,707277138662880504045,18974871706141125008640
%N A351737 Expansion of e.g.f. exp( x * (exp(3 * x) - 1) ).
%H A351737 Seiichi Manyama, <a href="/A351737/b351737.txt">Table of n, a(n) for n = 0..461</a>
%F A351737 a(n) = n! * Sum_{k=0..floor(n/2)} 3^(n-k) * Stirling2(n-k,k)/(n-k)!.
%F A351737 From _Seiichi Manyama_, Aug 29 2022: (Start)
%F A351737 a(n) = Sum_{k=0..n} (3*k-1)^(n-k) * binomial(n,k).
%F A351737 G.f.: Sum_{k>=0} x^k / (1 - (3*k-1)*x)^(k+1). (End)
%o A351737 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(exp(3*x)-1))))
%o A351737 (PARI) a(n) = n!*sum(k=0, n\2, 3^(n-k)*stirling(n-k, k, 2)/(n-k)!);
%o A351737 (PARI) a(n) = sum(k=0, n, (3*k-1)^(n-k)*binomial(n, k)); \\ _Seiichi Manyama_, Aug 29 2022
%o A351737 (PARI) my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-(3*k-1)*x)^(k+1))) \\ _Seiichi Manyama_, Aug 29 2022
%Y A351737 Cf. A052506, A351736.
%Y A351737 Cf. A351734, A351735.
%K A351737 nonn
%O A351737 0,3
%A A351737 _Seiichi Manyama_, May 20 2022