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

%I A261049 #13 Feb 18 2023 16:01:42
%S A261049 1,1,2,5,9,19,37,71,133,252,464,851,1547,2787,4985,8862,15639,27446,
%T A261049 47909,83168,143691,247109,423082,721360,1225119,2072762,3494359,
%U A261049 5870717,9830702,16409939,27309660,45316753,74986921,123748430,203686778,334421510,547735241
%N A261049 Expansion of Product_{k>=1} (1+x^k)^(p(k)), where p(k) is the partition function.
%C A261049 Number of strict multiset partitions of integer partitions of n. Weigh transform of A000041. - _Gus Wiseman_, Oct 11 2018
%H A261049 Alois P. Heinz, <a href="/A261049/b261049.txt">Table of n, a(n) for n = 0..1000</a>
%H A261049 R. Kaneiwa, <a href="http://projecteuclid.org/euclid.tjm/1270473566">An asymptotic formula for Cayley's double partition function p(2; n)</a>, Tokyo J. Math. 2, 137-158 (1979).
%e A261049 From _Gus Wiseman_, Oct 11 2018: (Start)
%e A261049 The a(1) = 1 through a(5) = 19 strict multiset partitions:
%e A261049   {{1}}  {{2}}    {{3}}        {{4}}          {{5}}
%e A261049          {{1,1}}  {{1,2}}      {{1,3}}        {{1,4}}
%e A261049                   {{1,1,1}}    {{2,2}}        {{2,3}}
%e A261049                   {{1},{2}}    {{1,1,2}}      {{1,1,3}}
%e A261049                   {{1},{1,1}}  {{1},{3}}      {{1,2,2}}
%e A261049                                {{1,1,1,1}}    {{1},{4}}
%e A261049                                {{1},{1,2}}    {{2},{3}}
%e A261049                                {{2},{1,1}}    {{1,1,1,2}}
%e A261049                                {{1},{1,1,1}}  {{1},{1,3}}
%e A261049                                               {{1},{2,2}}
%e A261049                                               {{2},{1,2}}
%e A261049                                               {{3},{1,1}}
%e A261049                                               {{1,1,1,1,1}}
%e A261049                                               {{1},{1,1,2}}
%e A261049                                               {{1,1},{1,2}}
%e A261049                                               {{2},{1,1,1}}
%e A261049                                               {{1},{1,1,1,1}}
%e A261049                                               {{1,1},{1,1,1}}
%e A261049                                               {{1},{2},{1,1}}
%e A261049 (End)
%p A261049 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p A261049       binomial(combinat[numbpart](i), j)*b(n-i*j, i-1), j=0..n/i)))
%p A261049     end:
%p A261049 a:= n-> b(n$2):
%p A261049 seq(a(n), n=0..40);  # _Alois P. Heinz_, Aug 08 2015
%t A261049 nmax=40; CoefficientList[Series[Product[(1+x^k)^PartitionsP[k],{k,1,nmax}],{x,0,nmax}],x]
%Y A261049 Cf. A000041, A001970, A026007, A027998, A248882, A102866, A256142.
%Y A261049 Cf. A047968, A050342, A089259, A305551, A320328, A320330, A320331.
%Y A261049 Row sums of A360742.
%K A261049 nonn
%O A261049 0,3
%A A261049 _Vaclav Kotesovec_, Aug 08 2015