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

4,10

Links

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

Formula

a(n) = [x^n y^4] 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), 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 *)

Crossrefs

Column k=4 of A319797.

Cf. A000217.

Sequence in context: A228570 A173305 A233867 * A220272 A298917 A322530

Adjacent sequences: A319811 A319812 A319813 * A319815 A319816 A319817

Keyword

nonn

Author

Alois P. Heinz, Sep 28 2018

Status

approved