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A318949
Number of ways to write n as an orderless product of orderless sums.
27
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
OFFSET
1,2
LINKS
FORMULA
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
EXAMPLE
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)
MATHEMATICA
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}]
PROG
(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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 05 2018
STATUS
approved