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A050342 Expansion of Product_{m>=1} (1+x^m)^A000009(m).

%I A050342 #27 Jun 25 2025 19:10:33
%S A050342 1,1,1,3,4,7,12,19,30,49,77,119,186,286,438,670,1014,1528,2300,3437,
%T A050342 5119,7603,11241,16564,24343,35650,52058,75820,110115,159510,230522,
%U A050342 332324,477994,686044,982519,1404243,2003063,2851720,4052429,5748440,8140007,11507125
%N A050342 Expansion of Product_{m>=1} (1+x^m)^A000009(m).
%C A050342 Number of partitions of n into distinct parts with one level of parentheses. Each "part" in parentheses is distinct from all others at the same level. Thus (2+1)+(1) is allowed but (2)+(1+1) and (2+1+1) are not.
%H A050342 Alois P. Heinz, <a href="/A050342/b050342.txt">Table of n, a(n) for n = 0..4000</a>
%H A050342 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A050342 Weigh transform of A000009.
%e A050342 4=(4)=(3)+(1)=(3+1)=(2+1)+(1).
%e A050342 From _Gus Wiseman_, Oct 11 2018: (Start)
%e A050342 a(n) is the number of set systems (sets of sets) whose multiset union is an integer partition of n. For example, the a(1) = 1 through a(6) = 12 set systems are:
%e A050342   {{1}}  {{2}}  {{3}}      {{4}}        {{5}}        {{6}}
%e A050342                 {{1,2}}    {{1,3}}      {{1,4}}      {{1,5}}
%e A050342                 {{1},{2}}  {{1},{3}}    {{2,3}}      {{2,4}}
%e A050342                            {{1},{1,2}}  {{1},{4}}    {{1,2,3}}
%e A050342                                         {{2},{3}}    {{1},{5}}
%e A050342                                         {{1},{1,3}}  {{2},{4}}
%e A050342                                         {{2},{1,2}}  {{1},{1,4}}
%e A050342                                                      {{1},{2,3}}
%e A050342                                                      {{2},{1,3}}
%e A050342                                                      {{3},{1,2}}
%e A050342                                                      {{1},{2},{3}}
%e A050342                                                      {{1},{2},{1,2}}
%e A050342 (End)
%p A050342 g:= proc(n, i) option remember; `if`(n=0, 1,
%p A050342       `if`(i<1, 0, g(n, i-1)+`if`(i>n, 0, g(n-i, i-1))))
%p A050342     end:
%p A050342 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A050342       add(binomial(g(i, i), j)*b(n-i*j, i-1), j=0..n/i)))
%p A050342     end:
%p A050342 a:= n-> b(n, n):
%p A050342 seq(a(n), n=0..50);  # _Alois P. Heinz_, May 19 2013
%t A050342 g[n_, i_] := g[n, i] = If[n==0, 1, If[i<1, 0, g[n, i-1] + If[i>n, 0, g[n-i, i-1]]]]; b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[Binomial[g[i, i], j]*b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Dec 19 2015, after _Alois P. Heinz_ *)
%t A050342 nn=10;Table[SeriesCoefficient[Product[(1+x^k)^PartitionsQ[k],{k,nn}],{x,0,n}],{n,0,nn}] (* _Gus Wiseman_, Oct 11 2018 *)
%Y A050342 Cf. A050343-A050350, A089254.
%Y A050342 Cf. A001970, A089259, A141268, A258466, A261049, A320328, A320330.
%Y A050342 Row sums of A330462 and of A360764.
%K A050342 nonn
%O A050342 0,4
%A A050342 _Christian G. Bower_, Oct 15 1999