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A325929 Total number of sub-subsets of set partitions of [n] where each subset is again partitioned into nonempty subsets.

%I A325929 #13 Dec 16 2020 15:13:17
%S A325929 0,1,4,14,57,262,1326,7499,47662,334794,2555639,21124116,189492474,
%T A325929 1838561337,19094196270,210014919406,2433655645025,29707254349866,
%U A325929 382324345380310,5179102279125987,73515985821539778,1087888385861343158,16724494503770495231
%N A325929 Total number of sub-subsets of set partitions of [n] where each subset is again partitioned into nonempty subsets.
%H A325929 Alois P. Heinz, <a href="/A325929/b325929.txt">Table of n, a(n) for n = 0..300</a>
%F A325929 a(n) = Sum_{k=1..n} k * A324162(n,k).
%p A325929 b:= proc(n, k) option remember; `if`(n=0, 1, `if`(k=0 or k>n, 0,
%p A325929       add(b(n-j, k)*binomial(n-1, j-1)*Stirling2(j, k), j=k..n)))
%p A325929     end:
%p A325929 a:= n-> add(b(n, k)*k, k=0..n):
%p A325929 seq(a(n), n=0..23);
%t A325929 b[n_, k_] := b[n, k] = If[n == 0, 1, If[k == 0 || k > n, 0, Sum[b[n - j, k] Binomial[n - 1, j - 1] StirlingS2[j, k], {j, k, n}]]];
%t A325929 a[n_] := Sum[b[n, k] k, {k, 0, n}];
%t A325929 a /@ Range[0, 23] (* _Jean-François Alcover_, Dec 16 2020, after _Alois P. Heinz_ *)
%Y A325929 Cf. A324162.
%K A325929 nonn
%O A325929 0,3
%A A325929 _Alois P. Heinz_, Sep 08 2019