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A212218 Number of representations of n as a sum of products of distinct pairs of positive integers, n = Sum_{k=1..m} i_k*j_k with i_k<=j_k, i_k

%I A212218 #14 Dec 05 2014 09:40:13
%S A212218 1,1,1,1,2,2,2,2,3,3,4,4,4,5,6,5,7,7,8,9,10,9,11,12,13,14,16,14,18,21,
%T A212218 19,20,23,23,28,28,28,30,36,33,39,42,39,44,50,46,54,57,56,62,69,64,71,
%U A212218 77,82,85,89,84,99,107,103,111,119,117,132,137,137,142
%N A212218 Number of representations of n as a sum of products of distinct pairs of positive integers, n = Sum_{k=1..m} i_k*j_k with i_k<=j_k, i_k<i_{k+1}, j_k<j_{k+1}.
%H A212218 Alois P. Heinz, <a href="/A212218/b212218.txt">Table of n, a(n) for n = 0..200</a>
%e A212218 a(0) = 1: 0 = the empty sum.
%e A212218 a(1) = 1: 1 = 1*1.
%e A212218 a(4) = 2: 4 = 1*4 = 2*2.
%e A212218 a(5) = 2: 5 = 1*1 + 2*2 = 1*5.
%e A212218 a(9) = 3: 9 = 1*1 + 2*4 = 1*9 = 3*3.
%e A212218 a(12) = 4: 12 = 1*2 + 2*5 = 1*12 = 2*6 = 3*4.
%e A212218 a(15) = 5: 15 = 1*3 + 2*6 = 1*3 + 3*4 = 1*1 + 2*7 = 1*15 = 3*5.
%p A212218 with(numtheory):
%p A212218 b:= proc(n, m, i, j) option remember;
%p A212218       `if`(n=0, 1, `if`(m<1, 0, b(n, m-1, i, j) +`if`(m>n, 0,
%p A212218         add(b(n-m, m-1, min(i, k-1), min(j, m/k-1)), k=select(x->
%p A212218          is(x<=min(sqrt(m), i) and m<=j*x), divisors(m))))))
%p A212218     end:
%p A212218 a:= n-> b(n$4):
%p A212218 seq(a(n), n=0..30);
%t A212218 b[n_, m_, i_, j_] := b[n, m, i, j] = If[n == 0, 1, If[m<1, 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], # <= Min[Sqrt[m], i] && m <= j*#&]}]]]]; a[n_] := b[n, n, n, n]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Dec 05 2014, after _Alois P. Heinz_ *)
%Y A212218 Cf. A066739, A182269, A182270, A211856, A211857, A212214, A212215, A212216, A212217, A212219.
%K A212218 nonn
%O A212218 0,5
%A A212218 _Alois P. Heinz_, May 06 2012