A319799 - OEIS

%I #15 Mar 12 2021 22:16:21

%S 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 17,17,21,20,23,24,27,28,31,32,36,37,42,43,47,50,53,58,61,64,69,72,82,

%U 83,91,92,102,107,115,118,128,135,147,152,159,169,181

%N Number of partitions of 2n into exactly n positive triangular numbers.

%H Alois P. Heinz, <a href="/A319799/b319799.txt">Table of n, a(n) for n = 0..3000</a>

%F a(n) = [x^(2n) y^n] 1/Product_{j>=1} (1-y*x^A000217(j)).

%F a(n) = A319797(2n,n).

%F G.f.: Product_{k>=1} 1 / (1 - x^(k*(k + 3)/2)). - _Ilya Gutkovskiy_, Jan 31 2021

%p h:= proc(n) option remember; `if`(n<1, 0,

%p       `if`(issqr(8*n+1), n, h(n-1)))

%p     end:

%p b:= proc(n, i, k) option remember; `if`(n=0, `if`(k=0, 1, 0),

%p       `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     end:

%p a:= n-> b(2*n, h(2*n), n):

%p seq(a(n), n=0..80);

%t h[n_] := h[n] = If[n < 1, 0, If[IntegerQ@Sqrt[8*n + 1], n, h[n - 1]]];

%t 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 a[n_] := b[2n, h[2n], n];

%t a /@ Range[0, 80] (* _Jean-François Alcover_, Mar 12 2021, after _Alois P. Heinz_ *)

%Y Cf. A000217, A111178, A319797.

%K nonn

%O 0,10

%A _Alois P. Heinz_, Sep 28 2018