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

Author

Vaclav Kotesovec, Aug 08 2015

Keywords

• nonn

Comments

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

Examples

			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)
		

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).

Crossrefs

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

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

Row sums of A360742.

Programs

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]