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A211857 Number of representations of n as a sum of products of distinct pairs of integers larger than 1, considered to be equivalent when terms or factors are reordered.

%I A211857 #26 Sep 23 2019 14:57:46
%S A211857 1,0,0,0,1,0,1,0,1,1,2,0,3,1,3,2,5,1,7,3,8,5,11,4,16,9,17,12,25,13,34,
%T A211857 20,37,28,53,32,69,46,78,63,108,71,136,100,160,134,210,152,265,211,
%U A211857 313,268,403,316,506,421,596,528,759,629,943,814,1111,1016
%N A211857 Number of representations of n as a sum of products of distinct pairs of integers larger than 1, considered to be equivalent when terms or factors are reordered.
%H A211857 Seiichi Manyama, <a href="/A211857/b211857.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)
%F A211857 G.f.: Product_{k>0} (1+x^k)^(A038548(k)-1). - _Vaclav Kotesovec_, Aug 19 2019
%F A211857 G.f.: Product_{i>=1} Product_{j=2..i} (1 + x^(i*j)). - _Ilya Gutkovskiy_, Sep 23 2019
%e A211857 a(0) = 1: 0 = the empty sum.
%e A211857 a(1) = a(2) = a(3) = 0: no product is < 4.
%e A211857 a(4) = 1: 4 = 2*2.
%e A211857 a(6) = 1: 6 = 2*3.
%e A211857 a(8) = 1: 8 = 2*4.
%e A211857 a(9) = 1: 9 = 3*3.
%e A211857 a(10) = 2: 10 = 2*2 + 2*3 = 2*5.
%e A211857 a(12) = 3: 12 = 2*2 + 2*4 = 2*6 = 3*4.
%e A211857 a(16) = 5: 16 = 2*2 + 2*6 = 2*2 + 3*4 = 2*3 + 2*5 = 2*8 = 4*4.
%p A211857 with(numtheory):
%p A211857 b:= proc(n, i) option remember; local c;
%p A211857       c:= ceil(tau(i)/2)-1;
%p A211857       `if`(n=0, 1, `if`(i<2, 0, b(n, i-1)
%p A211857        +add(b(n-i*j, i-1) *binomial(c, j), j=1..min(c, n/i))))
%p A211857     end:
%p A211857 a:= n-> b(n, n):
%p A211857 seq(a(n), n=0..70);
%t A211857 b[n_, i_] := b[n, i] = Module[{c}, c = Ceiling[DivisorSigma[0, i]/2]-1; If[n==0, 1, If[i<2, 0, b[n, i-1]+Sum[b[n-i*j, i-1]*Binomial[c, j], {j, 1, Min[c, n/i]}]]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 70}] (* _Jean-François Alcover_, Feb 19 2017, translated from Maple *)
%Y A211857 Cf. A000005, A006171, A038548, A066739, A182269, A182270, A211856.
%K A211857 nonn
%O A211857 0,11
%A A211857 _Alois P. Heinz_, Apr 22 2012