A261049 Expansion of Product_{k>=1} (1+x^k)^(p(k)), where p(k) is the partition function.

1, 1, 2, 5, 9, 19, 37, 71, 133, 252, 464, 851, 1547, 2787, 4985, 8862, 15639, 27446, 47909, 83168, 143691, 247109, 423082, 721360, 1225119, 2072762, 3494359, 5870717, 9830702, 16409939, 27309660, 45316753, 74986921, 123748430, 203686778, 334421510, 547735241

Offset

0,3

Comments

Number of strict multiset partitions of integer partitions of n. Weigh transform of A000041. - Gus Wiseman, Oct 11 2018

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

R. Kaneiwa, An asymptotic formula for Cayley's double partition function p(2; n), Tokyo J. Math. 2, 137-158 (1979).

Example

From Gus Wiseman, Oct 11 2018: (Start)
The a(1) = 1 through a(5) = 19 strict multiset partitions:
  {{1}}  {{2}}    {{3}}        {{4}}          {{5}}
         {{1,1}}  {{1,2}}      {{1,3}}        {{1,4}}
                  {{1,1,1}}    {{2,2}}        {{2,3}}
                  {{1},{2}}    {{1,1,2}}      {{1,1,3}}
                  {{1},{1,1}}  {{1},{3}}      {{1,2,2}}
                               {{1,1,1,1}}    {{1},{4}}
                               {{1},{1,2}}    {{2},{3}}
                               {{2},{1,1}}    {{1,1,1,2}}
                               {{1},{1,1,1}}  {{1},{1,3}}
                                              {{1},{2,2}}
                                              {{2},{1,2}}
                                              {{3},{1,1}}
                                              {{1,1,1,1,1}}
                                              {{1},{1,1,2}}
                                              {{1,1},{1,2}}
                                              {{2},{1,1,1}}
                                              {{1},{1,1,1,1}}
                                              {{1,1},{1,1,1}}
                                              {{1},{2},{1,1}}
(End)

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      binomial(combinat[numbpart](i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Aug 08 2015

Mathematica

nmax=40; CoefficientList[Series[Product[(1+x^k)^PartitionsP[k], {k, 1, nmax}], {x, 0, nmax}], x]

Crossrefs

Cf. A000041, A001970, A026007, A027998, A248882, A102866, A256142.

Cf. A047968, A050342, A089259, A305551, A320328, A320330, A320331.

Row sums of A360742.

Sequence in context: A014495 A056326 A280247 * A122893 A178841 A214319

Adjacent sequences: A261046 A261047 A261048 * A261050 A261051 A261052

Keyword

nonn

Author

Vaclav Kotesovec, Aug 08 2015

Status

approved