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%I #8 Oct 26 2018 20:36:36
%S 1,1,3,5,10,14,25,33,54,73,107,140,207,264,369,479,652,828,1112,1400,
%T 1848,2326,3009,3762,4856,6020,7648,9478,11942,14705,18427,22576,
%U 28083,34350,42429,51714,63680,77289,94618,114648,139773,168799,205144,247128,299310,359958,434443,521255,627812,751665,902862
%N Number of multiset partitions of integer partitions of n where all parts have the same product.
%e The a(1) = 1 through a(6) = 25 multiset partitions:
%e (1) (2) (3) (4) (5) (6)
%e (11) (12) (13) (14) (15)
%e (1)(1) (111) (22) (23) (24)
%e (1)(11) (112) (113) (33)
%e (1)(1)(1) (1111) (122) (114)
%e (2)(2) (1112) (123)
%e (1)(111) (11111) (222)
%e (11)(11) (2)(12) (1113)
%e (1)(1)(11) (1)(1111) (1122)
%e (1)(1)(1)(1) (11)(111) (3)(3)
%e (1)(1)(111) (11112)
%e (1)(11)(11) (111111)
%e (1)(1)(1)(11) (12)(12)
%e (1)(1)(1)(1)(1) (2)(112)
%e (2)(2)(2)
%e (1)(11111)
%e (11)(1111)
%e (111)(111)
%e (1)(1)(1111)
%e (1)(11)(111)
%e (11)(11)(11)
%e (1)(1)(1)(111)
%e (1)(1)(11)(11)
%e (1)(1)(1)(1)(11)
%e (1)(1)(1)(1)(1)(1)
%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t Table[Length[Select[Join@@mps/@IntegerPartitions[n],SameQ@@Times@@@#&]],{n,8}]
%o (PARI)
%o G(n)={my(M=Map()); for(k=1, n, forpart(p=k, my(t=vecprod(Vec(p)), z); mapput(M, t, if(mapisdefined(M, t, &z), z, 0) + x^k))); M}
%o a(n)=if(n==0, 1, vecsum(apply(p->EulerT(Vecrev(p/x, n))[n], Mat(G(n))[,2]))) \\ _Andrew Howroyd_, Oct 26 2018
%Y Cf. A001055, A001970, A045778, A050336, A279375, A294617, A294786, A294787, A294788, A320887, A320888, A320889.
%K nonn
%O 0,3
%A _Gus Wiseman_, Oct 23 2018
%E a(13)-a(50) from _Andrew Howroyd_, Oct 26 2018