cp's OEIS Frontend

A346920 Expansion of e.g.f. 1 / (1 - (exp(x) - 1)^5 / 5!).

%I A346920 #16 May 09 2022 15:18:58
%S A346920 1,0,0,0,0,1,15,140,1050,6951,42777,260590,1809060,17418401,229768539,
%T A346920 3402511476,50013258750,706670789371,9659104177101,130958047050698,
%U A346920 1834295186003784,27849428308615221,472297857494304303,8856291348143365456,176841068643273207426
%N A346920 Expansion of e.g.f. 1 / (1 - (exp(x) - 1)^5 / 5!).
%H A346920 Seiichi Manyama, <a href="/A346920/b346920.txt">Table of n, a(n) for n = 0..469</a>
%F A346920 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * Stirling2(k,5) * a(n-k).
%F A346920 a(n) ~ n! / (5*(1 + 120^(-1/5)) * log(1 + 120^(1/5))^(n+1)). - _Vaclav Kotesovec_, Aug 08 2021
%F A346920 From _Seiichi Manyama_, May 09 2022: (Start)
%F A346920 G.f.: Sum_{k>=0} (5*k)! * x^(5*k)/(120^k * Product_{j=1..5*k} (1 - j * x)).
%F A346920 a(n) = Sum_{k=0..floor(n/5)} (5*k)! * Stirling2(n,5*k)/120^k. (End)
%t A346920 nmax = 24; CoefficientList[Series[1/(1 - (Exp[x] - 1)^5/5!), {x, 0, nmax}], x] Range[0, nmax]!
%t A346920 a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] StirlingS2[k, 5] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 24}]
%o A346920 (PARI) my(x='x+O('x^25)); Vec(serlaplace(1/(1-(exp(x)-1)^5/5!))) \\ _Michel Marcus_, Aug 07 2021
%o A346920 (PARI) my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (5*k)!*x^(5*k)/(120^k*prod(j=1, 5*k, 1-j*x)))) \\ _Seiichi Manyama_, May 09 2022
%o A346920 (PARI) a(n) = sum(k=0, n\5, (5*k)!*stirling(n, 5*k, 2)/120^k); \\ _Seiichi Manyama_, May 09 2022
%Y A346920 Cf. A000670, A330047, A346894, A346895.
%Y A346920 Cf. A000481, A327506, A346924.
%K A346920 nonn
%O A346920 0,7
%A A346920 _Ilya Gutkovskiy_, Aug 07 2021