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A318949 Number of ways to write n as an orderless product of orderless sums.

%I A318949 #14 Oct 27 2019 14:08:23
%S A318949 1,2,3,8,7,17,15,36,36,56,56,123,101,165,197,310,297,490,490,767,837,
%T A318949 1114,1255,1925,1986,2638,3110,4108,4565,6201,6842,9043,10311,12904,
%U A318949 14988,19398,21637,26995,31488,39180,44583,55418,63261,77627,89914,108068,124754
%N A318949 Number of ways to write n as an orderless product of orderless sums.
%H A318949 Andrew Howroyd, <a href="/A318949/b318949.txt">Table of n, a(n) for n = 1..10000</a>
%F A318949 Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^p(k), where p(k) = number of partitions of k (A000041). - _Ilya Gutkovskiy_, Oct 26 2019
%e A318949 The a(6) = 17 ways:
%e A318949   (6)              (2)*(3)
%e A318949   (3+3)            (2)*(2+1)
%e A318949   (4+2)            (2)*(1+1+1)
%e A318949   (5+1)            (1+1)*(3)
%e A318949   (2+2+2)          (1+1)*(2+1)
%e A318949   (3+2+1)          (1+1)*(1+1+1)
%e A318949   (4+1+1)
%e A318949   (2+2+1+1)
%e A318949   (3+1+1+1)
%e A318949   (2+1+1+1+1)
%e A318949   (1+1+1+1+1+1)
%t A318949 facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
%t A318949 prodsums[n_]:=Union[Sort/@Join@@Table[Tuples[IntegerPartitions/@fac],{fac,facs[n]}]];
%t A318949 Table[Length[prodsums[n]],{n,30}]
%o A318949 (PARI) MultEulerT(u)={my(v=vector(#u)); v[1]=1; for(k=2, #u, forstep(j=#v\k*k, k, -k, my(i=j, e=0); while(i%k==0, i/=k; e++; v[j]+=binomial(e+u[k]-1, e)*v[i]))); v}
%o A318949 seq(n)={MultEulerT(vector(n, n, numbpart(n)))} \\ _Andrew Howroyd_, Oct 26 2019
%Y A318949 Cf. A000041, A001055, A001970, A063834, A065026, A066739, A066815, A281113, A284639, A318948.
%K A318949 nonn
%O A318949 1,2
%A A318949 _Gus Wiseman_, Sep 05 2018