A319814 Number of partitions of n into exactly four positive triangular numbers.

1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 2, 2, 1, 3, 2, 4, 2, 2, 3, 3, 3, 2, 4, 3, 5, 3, 2, 5, 4, 4, 3, 5, 4, 4, 5, 4, 5, 5, 4, 6, 5, 5, 6, 5, 5, 6, 7, 3, 5, 9, 5, 7, 5, 8, 7, 7, 4, 7, 9, 7, 8, 5, 7, 8, 10, 6, 6, 10, 7, 10, 7, 8, 9, 8, 8, 7, 13, 7, 10, 11

Offset: 4

Author

Alois P. Heinz, Sep 28 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 4..10000

Crossrefs

Column k=4 of A319797.

Cf. A000217.

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`(
      k>n or i*k b(n, h(n), 4):
seq(a(n), n=4..120);

Mathematica

h[n_] := h[n] = If[n<1, 0, If[IntegerQ@Sqrt[8n + 1], n, h[n - 1]]];
b[n_, i_, k_] := b[n, i, k] = If[n==0, If[k==0, 1, 0], If[k>n || i k < n, 0, b[n, h[i - 1], k] + b[n - i, h[Min[n - i, i]], k - 1]]];
a[n_] := b[n, h[n], 4];
a /@ Range[4, 120] (* Jean-François Alcover, Dec 13 2020, after Alois P. Heinz *)

Formula

a(n) = [x^n y^4] 1/Product_{j>=1} (1-y*x^A000217(j)).