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A212219 Number of representations of n as a sum of products of distinct pairs of positive integers >=2, n = Sum_{k=1..m} i_k*j_k with 2<=i_k<=j_k, i_k

%I A212219 #13 Dec 16 2014 13:56:20
%S A212219 1,0,0,0,1,0,1,0,1,1,1,0,2,1,1,1,3,0,3,1,3,2,3,1,5,3,3,2,6,4,5,3,6,6,
%T A212219 7,2,11,5,8,6,12,7,10,8,12,11,14,8,17,11,16,13,19,13,23,15,22,17,25,
%U A212219 18,29,24,24,23,36,25,34,25,42,34,39,30,47,40,48,37
%N A212219 Number of representations of n as a sum of products of distinct 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}.
%H A212219 Alois P. Heinz, <a href="/A212219/b212219.txt">Table of n, a(n) for n = 0..200</a>
%e A212219 a(0) = 1: 0 = the empty sum.
%e A212219 a(4) = 1: 4 = 2*2.
%e A212219 a(12) = 2: 12 = 2*6 = 3*4.
%e A212219 a(13) = 1: 13 = 2*2 + 3*3.
%e A212219 a(20) = 3: 20 = 2*2 + 4*4 = 2*10 = 4*5.
%e A212219 a(23) = 1: 23 = 2*4 + 3*5.
%e A212219 a(31) = 3: 31 = 2*5 + 3*7 = 2*3 + 5*5 = 2*2 + 3*9.
%p A212219 with(numtheory):
%p A212219 b:= proc(n, m, i, j) option remember;
%p A212219       `if`(n=0, 1, `if`(m<4, 0, b(n, m-1, i, j) +`if`(m>n, 0,
%p A212219         add(b(n-m, m-1, min(i, k-1), min(j, m/k-1)), k=select(x->
%p A212219          is(x>1 and x<=min(sqrt(m), i) and m<=j*x), divisors(m))))))
%p A212219     end:
%p A212219 a:= n-> b(n$4):
%p A212219 seq(a(n), n=0..30);
%t A212219 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-1, Min[i, k-1], Min[j, m/k-1]], {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_, Dec 09 2014, after _Alois P. Heinz_ *)
%Y A212219 Cf. A066739, A182269, A182270, A211856, A211857, A212214, A212215, A212216, A212217, A212218.
%K A212219 nonn
%O A212219 0,13
%A A212219 _Alois P. Heinz_, May 06 2012