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A066815 Number of partitions of n into sums of products.

%I A066815 #20 Nov 14 2018 08:13:20
%S A066815 1,1,2,3,6,8,14,19,33,45,69,94,148,197,289,390,575,762,1086,1439,2040,
%T A066815 2687,3712,4874,6749,8792,11918,15526,20998,27164,36277,46820,62367,
%U A066815 80146,105569,135326,177979,227139,296027,377142,490554,622526,804158
%N A066815 Number of partitions of n into sums of products.
%C A066815 Number of ways to choose a factorization of each part of an integer partition of n. - _Gus Wiseman_, Sep 05 2018
%C A066815 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
%H A066815 Seiichi Manyama, <a href="/A066815/b066815.txt">Table of n, a(n) for n = 0..10000</a>
%F A066815 G.f.: Product_{k>=1} 1/(1-A001055(k)*x^k).
%F A066815 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).
%e A066815 From _Gus Wiseman_, Sep 05 2018: (Start)
%e A066815 The a(6) = 14 partitions of 6 into sums of products:
%e A066815   6, 2*3,
%e A066815   5+1, 4+2, 2*2+2, 3+3,
%e A066815   4+1+1, 2*2+1+1, 3+2+1, 2+2+2,
%e A066815   3+1+1+1, 2+2+1+1,
%e A066815   2+1+1+1+1,
%e A066815   1+1+1+1+1+1.
%e A066815 (End)
%t A066815 facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
%t A066815 Table[Length[Join@@Table[Tuples[facs/@ptn],{ptn,IntegerPartitions[n]}]],{n,20}] (* _Gus Wiseman_, Sep 05 2018 *)
%Y A066815 Cf. A000041, A001055, A066739, A321460.
%Y A066815 Cf. A001970, A050336, A063834, A065026, A281113, A284639, A318948, A318949.
%K A066815 nonn
%O A066815 0,3
%A A066815 _Vladeta Jovovic_, Jan 20 2002
%E A066815 Renamed by _T. D. Noe_, May 24 2011