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A066815
Number of partitions of n into sums of products.
15
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
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 *)
KEYWORD
nonn
AUTHOR
Vladeta Jovovic, Jan 20 2002
EXTENSIONS
Renamed by T. D. Noe, May 24 2011
STATUS
approved