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A212217 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}, i_k*j_k

%I A212217 #12 Feb 10 2017 09:10:24
%S A212217 1,0,0,0,1,0,1,0,1,1,2,0,3,1,3,2,5,0,7,2,8,3,10,1,15,6,14,6,21,6,28,9,
%T A212217 26,14,38,12,50,16,47,26,70,19,82,31,87,47,111,33,141,58,143,71,182,
%U A212217 63,228,93,231,117,289,102,364,148,354,187,462,172,537,227
%N A212217 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}, i_k*j_k<i_{k+1}*j_{k+1}.
%H A212217 Alois P. Heinz, <a href="/A212217/b212217.txt">Table of n, a(n) for n = 0..200</a>
%e A212217 a(0) = 1: 0 = the empty sum.
%e A212217 a(4) = 1: 4 = 2*2.
%e A212217 a(6) = 1: 6 = 2*3.
%e A212217 a(8) = 1: 8 = 2*4.
%e A212217 a(9) = 1: 9 = 3*3.
%e A212217 a(10) = 2: 10 = 2*2 + 2*3 = 2*5.
%e A212217 a(12) = 3: 12 = 2*2 + 2*4 = 2*6 = 3*4.
%e A212217 a(13) = 1: 13 = 2*2 + 3*3.
%e A212217 a(14) = 3: 14 = 2*3 + 2*4 = 2*2 + 2*5 = 2*7.
%e A212217 a(15) = 2: 15 = 2*3 + 3*3 = 3*5.
%e A212217 a(16) = 5: 16 = 2*3 + 2*5 = 2*2 + 2*6 = 2*2 + 3*4 = 2*8 = 4*4.
%e A212217 a(19) = 2: 19 = 2*2 + 2*3 + 3*3 = 2*2 + 3*5.
%p A212217 with(numtheory):
%p A212217 b:= proc(n, m, i, j) option remember;
%p A212217       `if`(n=0, 1, `if`(m<4, 0, b(n, m-1, i, j) +`if`(m>n, 0,
%p A212217         add(b(n-m, m-1, min(i, k), min(j, m/k)), k=select(x->
%p A212217          is(x>1 and x<=min(sqrt(m), i) and m<=j*x), divisors(m))))))
%p A212217     end:
%p A212217 a:= n-> b(n$4):
%p A212217 seq(a(n), n=0..30);
%t A212217 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], Min[j, m/k]], {k, Select[Divisors[m], # > 1 && # <= Min [Sqrt[m], i] && m <= j*# &]}]]]];
%t A212217 a[n_] := b[n, n, n, n]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jan 23 2017, after _Alois P. Heinz_ *)
%Y A212217 Cf. A066739, A182269, A182270, A211856, A211857, A212214, A212215, A212216, A212218, A212219.
%K A212217 nonn
%O A212217 0,11
%A A212217 _Alois P. Heinz_, May 06 2012