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A319799 Number of partitions of 2n into exactly n positive triangular numbers.

%I A319799 #15 Mar 12 2021 22:16:21
%S A319799 1,0,1,0,1,1,1,1,1,2,2,2,2,2,4,3,4,3,5,5,7,5,7,7,9,9,9,11,12,14,14,14,
%T A319799 17,17,21,20,23,24,27,28,31,32,36,37,42,43,47,50,53,58,61,64,69,72,82,
%U A319799 83,91,92,102,107,115,118,128,135,147,152,159,169,181
%N A319799 Number of partitions of 2n into exactly n positive triangular numbers.
%H A319799 Alois P. Heinz, <a href="/A319799/b319799.txt">Table of n, a(n) for n = 0..3000</a>
%F A319799 a(n) = [x^(2n) y^n] 1/Product_{j>=1} (1-y*x^A000217(j)).
%F A319799 a(n) = A319797(2n,n).
%F A319799 G.f.: Product_{k>=1} 1 / (1 - x^(k*(k + 3)/2)). - _Ilya Gutkovskiy_, Jan 31 2021
%p A319799 h:= proc(n) option remember; `if`(n<1, 0,
%p A319799       `if`(issqr(8*n+1), n, h(n-1)))
%p A319799     end:
%p A319799 b:= proc(n, i, k) option remember; `if`(n=0, `if`(k=0, 1, 0),
%p A319799       `if`(i*k<n or k>n, 0, b(n, h(i-1), k)+b(n-i, h(min(n-i, i)), k-1)))
%p A319799     end:
%p A319799 a:= n-> b(2*n, h(2*n), n):
%p A319799 seq(a(n), n=0..80);
%t A319799 h[n_] := h[n] = If[n < 1, 0, If[IntegerQ@Sqrt[8*n + 1], n, h[n - 1]]];
%t A319799 b[n_, i_, k_] := b[n, i, k] = If[n == 0, If[k == 0, 1, 0], If[i*k < n || k > n, 0, b[n, h[i - 1], k] + b[n - i, h[Min[n - i, i]], k - 1]]];
%t A319799 a[n_] := b[2n, h[2n], n];
%t A319799 a /@ Range[0, 80] (* _Jean-François Alcover_, Mar 12 2021, after _Alois P. Heinz_ *)
%Y A319799 Cf. A000217, A111178, A319797.
%K A319799 nonn
%O A319799 0,10
%A A319799 _Alois P. Heinz_, Sep 28 2018