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A327504 Number of set partitions of [n] where each subset is again partitioned into three nonempty subsets.

%I A327504 #14 May 08 2022 08:24:35
%S A327504 1,0,0,1,6,25,100,511,3626,29765,250200,2146771,19575446,195336505,
%T A327504 2124840900,24646324431,299803782466,3809251939245,50698296967600,
%U A327504 708349718638891,10372758309704686,158546862369781985,2519789706502636700,41545703617137280551
%N A327504 Number of set partitions of [n] where each subset is again partitioned into three nonempty subsets.
%H A327504 Alois P. Heinz, <a href="/A327504/b327504.txt">Table of n, a(n) for n = 0..501</a>
%H A327504 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A327504 E.g.f.: exp((exp(x)-1)^3/3!).
%F A327504 a(n) = Sum_{k=0..floor(n/3)} (3*k)! * Stirling2(n,3*k)/(6^k * k!). - _Seiichi Manyama_, May 07 2022
%p A327504 a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)
%p A327504       *binomial(n-1, j-1)*Stirling2(j, 3), j=3..n))
%p A327504     end:
%p A327504 seq(a(n), n=0..25);
%t A327504 a[n_] := a[n] = If[n == 0, 1, Sum[a[n - j] Binomial[n - 1, j -1] StirlingS2[j, 3], {j, 3, n}]];
%t A327504 a /@ Range[0, 25] (* _Jean-François Alcover_, Dec 16 2020, after _Alois P. Heinz_ *)
%o A327504 (PARI) a(n) = sum(k=0, n\3, (3*k)!*stirling(n, 3*k, 2)/(6^k*k!)); \\ _Seiichi Manyama_, May 07 2022
%Y A327504 Column k=3 of A324162.
%Y A327504 Cf. A346894.
%K A327504 nonn
%O A327504 0,5
%A A327504 _Alois P. Heinz_, Sep 14 2019