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A346894 Expansion of e.g.f. 1 / (1 - (exp(x) - 1)^3 / 3!).

%I A346894 #29 May 09 2022 08:35:53
%S A346894 1,0,0,1,6,25,110,721,6286,57625,541470,5558641,64351166,819480025,
%T A346894 11140978030,160711583761,2472834185646,40597082635225,
%U A346894 706816137889790,12974021811748081,250395124862965726,5074637684604691225,107798916619788396750
%N A346894 Expansion of e.g.f. 1 / (1 - (exp(x) - 1)^3 / 3!).
%H A346894 Seiichi Manyama, <a href="/A346894/b346894.txt">Table of n, a(n) for n = 0..452</a>
%F A346894 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * Stirling2(k,3) * a(n-k).
%F A346894 a(n) ~ n! / (3*(1 + 6^(-1/3)) * log(1 + 6^(1/3))^(n+1)). - _Vaclav Kotesovec_, Aug 08 2021
%F A346894 From _Seiichi Manyama_, May 07 2022: (Start)
%F A346894 G.f.: Sum_{k>=0} (3*k)! * x^(3*k)/(6^k * Product_{j=1..3*k} (1 - j * x)).
%F A346894 a(n) = Sum_{k=0..floor(n/3)} (3*k)! * Stirling2(n,3*k)/6^k. (End)
%t A346894 nmax = 22; CoefficientList[Series[1/(1 - (Exp[x] - 1)^3/3!), {x, 0, nmax}], x] Range[0, nmax]!
%t A346894 a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] StirlingS2[k, 3] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]
%o A346894 (PARI) my(x='x+O('x^25)); Vec(serlaplace(1/(1-(exp(x)-1)^3/3!))) \\ _Michel Marcus_, Aug 06 2021
%o A346894 (PARI) my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (3*k)!*x^(3*k)/(6^k*prod(j=1, 3*k, 1-j*x)))) \\ _Seiichi Manyama_, May 07 2022
%o A346894 (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, binomial(i, j)*stirling(j, 3, 2)*v[i-j+1])); v; \\ _Seiichi Manyama_, May 07 2022
%o A346894 (PARI) a(n) = sum(k=0, n\3, (3*k)!*stirling(n, 3*k, 2)/6^k); \\ _Seiichi Manyama_, May 07 2022
%Y A346894 Cf. A000670, A330047, A346895, A346920.
%Y A346894 Cf. A000392, A327504, A346922, A353774.
%K A346894 nonn
%O A346894 0,5
%A A346894 _Ilya Gutkovskiy_, Aug 06 2021