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A318949
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Number of ways to write n as an orderless product of orderless sums.
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27
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1, 2, 3, 8, 7, 17, 15, 36, 36, 56, 56, 123, 101, 165, 197, 310, 297, 490, 490, 767, 837, 1114, 1255, 1925, 1986, 2638, 3110, 4108, 4565, 6201, 6842, 9043, 10311, 12904, 14988, 19398, 21637, 26995, 31488, 39180, 44583, 55418, 63261, 77627, 89914, 108068, 124754
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OFFSET
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1,2
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LINKS
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FORMULA
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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
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EXAMPLE
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The a(6) = 17 ways:
(6) (2)*(3)
(3+3) (2)*(2+1)
(4+2) (2)*(1+1+1)
(5+1) (1+1)*(3)
(2+2+2) (1+1)*(2+1)
(3+2+1) (1+1)*(1+1+1)
(4+1+1)
(2+2+1+1)
(3+1+1+1)
(2+1+1+1+1)
(1+1+1+1+1+1)
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MATHEMATICA
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facs[n_]:=If[n<=1, {{}}, Join@@Table[(Prepend[#1, d]&)/@Select[facs[n/d], Min@@#1>=d&], {d, Rest[Divisors[n]]}]];
prodsums[n_]:=Union[Sort/@Join@@Table[Tuples[IntegerPartitions/@fac], {fac, facs[n]}]];
Table[Length[prodsums[n]], {n, 30}]
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PROG
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(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}
seq(n)={MultEulerT(vector(n, n, numbpart(n)))} \\ Andrew Howroyd, Oct 26 2019
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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