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

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

Author

Gus Wiseman, Sep 05 2018

Keywords

• nonn

Examples

			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)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000041, A001055, A001970, A063834, A065026, A066739, A066815, A281113, A284639, A318948.

Programs

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}]

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

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