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

0, 1, 4, 14, 57, 262, 1326, 7499, 47662, 334794, 2555639, 21124116, 189492474, 1838561337, 19094196270, 210014919406, 2433655645025, 29707254349866, 382324345380310, 5179102279125987, 73515985821539778, 1087888385861343158, 16724494503770495231

Offset: 0

Author

Alois P. Heinz, Sep 08 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A324162.

Programs

Maple

b:= proc(n, k) option remember; `if`(n=0, 1, `if`(k=0 or k>n, 0,
      add(b(n-j, k)*binomial(n-1, j-1)*Stirling2(j, k), j=k..n)))
    end:
a:= n-> add(b(n, k)*k, k=0..n):
seq(a(n), n=0..23);

Mathematica

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}]]];
a[n_] := Sum[b[n, k] k, {k, 0, n}];
a /@ Range[0, 23] (* Jean-François Alcover, Dec 16 2020, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=1..n} k * A324162(n,k).