A326346 Total number of partitions in the partitions of compositions of n.

0, 1, 4, 14, 47, 151, 474, 1457, 4414, 13210, 39155, 115120, 336183, 976070, 2819785, 8110657, 23239662, 66362960, 188930728, 536407146, 1519205230, 4293061640, 12106883585, 34079016842, 95762829405, 268670620736, 752676269695, 2105751165046, 5883798478398

Offset: 0

Author

Alois P. Heinz, Sep 11 2019

Keywords

• nonn

Examples

			a(3) = 14 = 1+1+1+2+2+2+2+3 counts the partitions in 3, 21, 111, 2|1, 11|1, 1|2, 1|11, 1|1|1.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2313

Crossrefs

Cf. A000041, A030267, A055887, A060642, A304969, A325915, A327548.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, [1, 0], (p-> p+
      [0, p[1]])(add(combinat[numbpart](j)*b(n-j), j=1..n)))
    end:
a:= n-> b(n)[2]:
seq(a(n), n=0..32);

Mathematica

b[n_] := b[n] = If[n==0, {1, 0}, Function[p, p + {0, p[[1]]}][Sum[ PartitionsP[j] b[n-j], {j, 1, n}]]];
a[n_] := b[n][[2]];
a /@ Range[0, 32] (* Jean-François Alcover, Dec 05 2020, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=1..n} k * A060642(n,k).

a(n) ~ c * d^n * n, where d = A246828 = 2.69832910647421123126399866618837633... and c = 0.171490233695958246364725709205670983251448838158816... - Vaclav Kotesovec, Sep 14 2019