A319815 Number of partitions of n into exactly five positive triangular numbers.

1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 3, 2, 3, 2, 3, 3, 5, 2, 4, 4, 4, 5, 3, 5, 5, 7, 4, 5, 6, 5, 8, 6, 6, 7, 8, 6, 7, 9, 8, 8, 9, 8, 10, 10, 8, 11, 10, 9, 10, 11, 11, 12, 14, 8, 13, 14, 13, 11, 13, 14, 16, 15, 10, 16, 16, 15, 15, 16, 16, 16, 21, 12, 18

Offset

5,10

Links

Alois P. Heinz, Table of n, a(n) for n = 5..10000

Formula

a(n) = [x^n y^5] 1/Product_{j>=1} (1-y*x^A000217(j)).

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<n, 0, b(n, h(i-1), k)+b(n-i, h(min(n-i, i)), k-1)))
    end:
a:= n-> b(n, h(n), 5):
seq(a(n), n=5..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], 5];
a /@ Range[5, 120] (* Jean-François Alcover, Dec 13 2020, after Alois P. Heinz *)

Crossrefs

Column k=5 of A319797.

Cf. A000217.

Sequence in context: A194314 A006371 A000177 * A222656 A162545 A162544

Adjacent sequences: A319812 A319813 A319814 * A319816 A319817 A319818

Keyword

nonn

Author

Alois P. Heinz, Sep 28 2018

Status

approved