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A327506 Number of set partitions of [n] where each subset is again partitioned into five nonempty subsets.

%I A327506 #11 May 08 2022 08:24:26
%S A327506 1,0,0,0,0,1,15,140,1050,6951,42651,253660,1594230,12463451,134921787,
%T A327506 1806386946,25524454410,354189159871,4751404201923,62042283083648,
%U A327506 803415873180606,10624141898153091,148849893975447819,2279247411153872566,38395707003954897234
%N A327506 Number of set partitions of [n] where each subset is again partitioned into five nonempty subsets.
%H A327506 Alois P. Heinz, <a href="/A327506/b327506.txt">Table of n, a(n) for n = 0..494</a>
%H A327506 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A327506 E.g.f.: exp((exp(x)-1)^5/5!).
%F A327506 a(n) = Sum_{k=0..floor(n/5)} (5*k)! * Stirling2(n,5*k)/(120^k * k!). - _Seiichi Manyama_, May 07 2022
%p A327506 a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)
%p A327506       *binomial(n-1, j-1)*Stirling2(j, 5), j=5..n))
%p A327506     end:
%p A327506 seq(a(n), n=0..25);
%t A327506 a[n_] := a[n] = If[n == 0, 1, Sum[a[n - j] Binomial[n - 1, j - 1] StirlingS2[j, 5], {j, 5, n}]];
%t A327506 a /@ Range[0, 25] (* _Jean-François Alcover_, Dec 16 2020, after _Alois P. Heinz_ *)
%o A327506 (PARI) a(n) = sum(k=0, n\5, (5*k)!*stirling(n, 5*k, 2)/(120^k*k!)); \\ _Seiichi Manyama_, May 07 2022
%Y A327506 Column k=5 of A324162.
%K A327506 nonn
%O A327506 0,7
%A A327506 _Alois P. Heinz_, Sep 14 2019