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A212215 Number of representations of n as a sum of products of pairs of positive integers >=2, n = Sum_{k=1..m} i_k*j_k with 2<=i_k<=j_k, i_k<=i_{k+1}, j_k<=j_{k+1}, i_k*j_k<=i_{k+1}*j_{k+1}.

%I A212215 #11 Jan 23 2017 08:13:22
%S A212215 1,0,0,0,1,0,1,0,2,1,2,0,5,1,4,2,9,1,11,2,16,5,18,3,33,8,31,11,52,11,
%T A212215 64,16,83,29,100,26,152,39,159,59,233,61,280,83,354,129,423,122,591,
%U A212215 180,644,241,864,260,1050,341,1282,472,1523,490,2016,655,2224
%N A212215 Number of representations of n as a sum of products of pairs of positive integers >=2, n = Sum_{k=1..m} i_k*j_k with 2<=i_k<=j_k, i_k<=i_{k+1}, j_k<=j_{k+1}, i_k*j_k<=i_{k+1}*j_{k+1}.
%H A212215 Alois P. Heinz, <a href="/A212215/b212215.txt">Table of n, a(n) for n = 0..200</a>
%e A212215 a(0) = 1: 0 = the empty sum.
%e A212215 a(4) = 1: 4 = 2*2.
%e A212215 a(6) = 1: 6 = 2*3.
%e A212215 a(8) = 2: 8 = 2*2 + 2*2 = 2*4.
%e A212215 a(9) = 1: 9 = 3*3.
%e A212215 a(10) = 2: 10 = 2*2 + 2*3 = 2*5.
%e A212215 a(17) = 1 = A182270(17)-1, one of the 2 sums counted by A182270(17) is not allowed: 17 = 2*4 + 3*3.
%p A212215 with(numtheory):
%p A212215 b:= proc(n, m, i, j) option remember;
%p A212215       `if`(n=0, 1, `if`(m<4, 0, b(n, m-1, i, j) +`if`(m>n, 0,
%p A212215         add(b(n-m, m, min(i, k), min(j, m/k)), k=select(x->
%p A212215          is(x>1 and x<=min(sqrt(m), i) and m<=j*x), divisors(m))))))
%p A212215     end:
%p A212215 a:= n-> b(n$4):
%p A212215 seq(a(n), n=0..30);
%t A212215 b[n_, m_, i_, j_] := b[n, m, i, j] = If[n == 0, 1, If[m < 4, 0, b[n, m - 1, i, j] + If[m > n, 0, Sum[b[n - m, m, Min[i, k], Min[j, m/k]], {k, Select[ Divisors[m], # > 1 && # <= Min[Sqrt[m], i] && m <= j*# &]}]]]]; a[n_] := b[n, n, n, n]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jan 23 2017, after _Alois P. Heinz_ *)
%Y A212215 Cf. A066739, A182269, A182270, A211856, A211857, A212214, A212216, A212217, A212218, A212219.
%K A212215 nonn
%O A212215 0,9
%A A212215 _Alois P. Heinz_, May 06 2012