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A346895 Expansion of e.g.f. 1 / (1 - (exp(x) - 1)^4 / 4!).

%I A346895 #23 May 09 2022 07:34:22
%S A346895 1,0,0,0,1,10,65,350,1771,10290,86605,977350,11778041,138208070,
%T A346895 1590920695,18895490250,245692484311,3587464083850,57397496312585,
%U A346895 966066470023550,16713560617838581,297182550111615630,5500448659383161275,107267326981597659250
%N A346895 Expansion of e.g.f. 1 / (1 - (exp(x) - 1)^4 / 4!).
%H A346895 Seiichi Manyama, <a href="/A346895/b346895.txt">Table of n, a(n) for n = 0..461</a>
%F A346895 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * Stirling2(k,4) * a(n-k).
%F A346895 a(n) ~ n! / (4*(1 + 2^(-3/4)*3^(-1/4)) * log(1 + 2^(3/4)*3^(1/4))^(n+1)). - _Vaclav Kotesovec_, Aug 08 2021
%F A346895 From _Seiichi Manyama_, May 07 2022: (Start)
%F A346895 G.f.: Sum_{k>=0} (4*k)! * x^(4*k)/(24^k * Product_{j=1..4*k} (1 - j * x)).
%F A346895 a(n) = Sum_{k=0..floor(n/4)} (4*k)! * Stirling2(n,4*k)/24^k. (End)
%t A346895 nmax = 23; CoefficientList[Series[1/(1 - (Exp[x] - 1)^4/4!), {x, 0, nmax}], x] Range[0, nmax]!
%t A346895 a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] StirlingS2[k, 4] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]
%o A346895 (PARI) my(x='x+O('x^25)); Vec(serlaplace(1/(1-(exp(x)-1)^4/4!))) \\ _Michel Marcus_, Aug 06 2021
%o A346895 (PARI) my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (4*k)!*x^(4*k)/(24^k*prod(j=1, 4*k, 1-j*x)))) \\ _Seiichi Manyama_, May 07 2022
%o A346895 (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, 4, 2)*v[i-j+1])); v; \\ _Seiichi Manyama_, May 07 2022
%o A346895 (PARI) a(n) = sum(k=0, n\4, (4*k)!*stirling(n, 4*k, 2)/24^k); \\ _Seiichi Manyama_, May 07 2022
%Y A346895 Cf. A000670, A330047, A346894, A346920.
%Y A346895 Cf. A000453, A327505, A346923, A353775.
%K A346895 nonn
%O A346895 0,6
%A A346895 _Ilya Gutkovskiy_, Aug 06 2021