A066815 Number of partitions of n into sums of products.

1, 1, 2, 3, 6, 8, 14, 19, 33, 45, 69, 94, 148, 197, 289, 390, 575, 762, 1086, 1439, 2040, 2687, 3712, 4874, 6749, 8792, 11918, 15526, 20998, 27164, 36277, 46820, 62367, 80146, 105569, 135326, 177979, 227139, 296027, 377142, 490554, 622526, 804158

Offset

0,3

Comments

Number of ways to choose a factorization of each part of an integer partition of n. - Gus Wiseman, Sep 05 2018

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1, g(n) = A001055(n). - Seiichi Manyama, Nov 14 2018

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Formula

G.f.: Product_{k>=1} 1/(1-A001055(k)*x^k).

a(n) = 1/n*Sum_{k=1..n} a(n-k)*b(k), n > 0, a(0)=1, b(k)=Sum_{d|k} d*(A001055(d))^(k/d).

Example

From Gus Wiseman, Sep 05 2018: (Start)
The a(6) = 14 partitions of 6 into sums of products:
  6, 2*3,
  5+1, 4+2, 2*2+2, 3+3,
  4+1+1, 2*2+1+1, 3+2+1, 2+2+2,
  3+1+1+1, 2+2+1+1,
  2+1+1+1+1,
  1+1+1+1+1+1.
(End)

Mathematica

facs[n_]:=If[n<=1, {{}}, Join@@Table[(Prepend[#1, d]&)/@Select[facs[n/d], Min@@#1>=d&], {d, Rest[Divisors[n]]}]];
Table[Length[Join@@Table[Tuples[facs/@ptn], {ptn, IntegerPartitions[n]}]], {n, 20}] (* Gus Wiseman, Sep 05 2018 *)

Crossrefs

Cf. A000041, A001055, A066739, A321460.

Cf. A001970, A050336, A063834, A065026, A281113, A284639, A318948, A318949.

Sequence in context: A321360 A321566 A066739 * A106182 A360621 A097097

Adjacent sequences: A066812 A066813 A066814 * A066816 A066817 A066818

Keyword

nonn

Author

Vladeta Jovovic, Jan 20 2002

Extensions

Renamed by T. D. Noe, May 24 2011

Status

approved