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A327505 Number of set partitions of [n] where each subset is again partitioned into four nonempty subsets.

%I A327505 #13 May 08 2022 08:24:31
%S A327505 1,0,0,0,1,10,65,350,1736,9030,60355,561550,6183221,69469400,
%T A327505 761767370,8239194600,91058524831,1073790441370,13900626022985,
%U A327505 196759304278250,2963381404815566,46227649788125190,736940002561065325,12005645243802471250,201482801573414254301
%N A327505 Number of set partitions of [n] where each subset is again partitioned into four nonempty subsets.
%H A327505 Alois P. Heinz, <a href="/A327505/b327505.txt">Table of n, a(n) for n = 0..495</a>
%H A327505 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A327505 E.g.f.: exp((exp(x)-1)^4/4!).
%F A327505 a(n) = Sum_{k=0..floor(n/4)} (4*k)! * Stirling2(n,4*k)/(24^k * k!). - _Seiichi Manyama_, May 07 2022
%p A327505 a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)
%p A327505       *binomial(n-1, j-1)*Stirling2(j, 4), j=4..n))
%p A327505     end:
%p A327505 seq(a(n), n=0..25);
%t A327505 a[n_] := a[n] = If[n == 0, 1, Sum[a[n - j] Binomial[n - 1, j - 1] StirlingS2[j, 4], {j, 4, n}]];
%t A327505 a /@ Range[0, 25] (* _Jean-François Alcover_, Dec 16 2020, after _Alois P. Heinz_ *)
%o A327505 (PARI) a(n) = sum(k=0, n\4, (4*k)!*stirling(n, 4*k, 2)/(24^k*k!)); \\ _Seiichi Manyama_, May 07 2022
%Y A327505 Column k=4 of A324162.
%Y A327505 Cf. A346895.
%K A327505 nonn
%O A327505 0,6
%A A327505 _Alois P. Heinz_, Sep 14 2019