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A212216 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}, i_k*j_k

%I A212216 #13 Dec 04 2014 04:14:08
%S A212216 1,1,1,2,3,4,6,7,8,12,15,18,25,28,34,44,51,59,75,87,103,124,143,163,
%T A212216 198,228,261,310,354,404,479,538,612,708,802,907,1049,1175,1320,1518,
%U A212216 1711,1910,2187,2431,2724,3097,3447,3843,4348,4818,5373,6032,6693,7420
%N A212216 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}, i_k*j_k<i_{k+1}*j_{k+1}.
%H A212216 Alois P. Heinz, <a href="/A212216/b212216.txt">Table of n, a(n) for n = 0..200</a>
%e A212216 a(0) = 1: 0 = the empty sum.
%e A212216 a(1) = 1: 1 = 1*1.
%e A212216 a(2) = 1: 2 = 1*2.
%e A212216 a(3) = 2: 3 = 1*1 + 1*2 = 1*3.
%e A212216 a(4) = 3: 4 = 1*1 + 1*3 = 1*4 = 2*2.
%e A212216 a(5) = 4: 5 = 1*2 + 1*3 = 1*1 + 1*4 = 1*1 + 2*2 = 1*5.
%e A212216 a(6) = 6: 6 = 1*1 + 1*2 + 1*3 = 1*2 + 1*4 = 1*2 + 2*2 = 1*1 + 1+5 = 1*6 = 2*3.
%p A212216 with(numtheory):
%p A212216 b:= proc(n, m, i, j) option remember;
%p A212216       `if`(n=0, 1, `if`(m<1, 0, b(n, m-1, i, j) +`if`(m>n, 0,
%p A212216         add(b(n-m, m-1, min(i, k), min(j, m/k)), k=select(x->
%p A212216          is(x<=min(sqrt(m), i) and m<=j*x), divisors(m))))))
%p A212216     end:
%p A212216 a:= n-> b(n$4):
%p A212216 seq(a(n), n=0..30);
%t A212216 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], Min[j, m/k]], {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 04 2014, after _Alois P. Heinz_ *)
%Y A212216 Cf. A066739, A182269, A182270, A211856, A211857, A212214, A212215, A212217, A212218, A212219.
%K A212216 nonn
%O A212216 0,4
%A A212216 _Alois P. Heinz_, May 06 2012