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

0,3

Links

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

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

Example

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.

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_ *)

Crossrefs

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

Sequence in context: A362279 A121530 A121299 * A046718 A291385 A192877

Adjacent sequences: A326343 A326344 A326345 * A326347 A326348 A326349

Keyword

nonn

Author

_Alois P. Heinz_, Sep 11 2019

Status

approved