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

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, 17, 17, 21, 20, 23, 24, 27, 28, 31, 32, 36, 37, 42, 43, 47, 50, 53, 58, 61, 64, 69, 72, 82, 83, 91, 92, 102, 107, 115, 118, 128, 135, 147, 152, 159, 169, 181

Offset: 0

Author

Alois P. Heinz, Sep 28 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..3000

Crossrefs

Cf. A000217, A111178, A319797.

Programs

Maple

h:= proc(n) option remember; `if`(n<1, 0,
      `if`(issqr(8*n+1), n, h(n-1)))
    end:
b:= proc(n, i, k) option remember; `if`(n=0, `if`(k=0, 1, 0),
      `if`(i*kn, 0, b(n, h(i-1), k)+b(n-i, h(min(n-i, i)), k-1)))
    end:
a:= n-> b(2*n, h(2*n), n):
seq(a(n), n=0..80);

Mathematica

h[n_] := h[n] = If[n < 1, 0, If[IntegerQ@Sqrt[8*n + 1], n, h[n - 1]]];
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]]];
a[n_] := b[2n, h[2n], n];
a /@ Range[0, 80] (* Jean-François Alcover, Mar 12 2021, after Alois P. Heinz *)

Formula

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

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

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