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

0,10

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..3000 from Alois P. Heinz)

Vaclav Kotesovec, Graph - the asymptotic ratio (500000 terms)

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

a(n) ~ sqrt(3) * zeta(3/2)^(5/3) * exp(3*Pi^(1/3)*zeta(3/2)^(2/3)*n^(1/3)/2) / (2^(9/2) * Pi^(2/3) * n^(13/6)) * (1 + (9*Pi^(2/3)*zeta(1/2)*zeta(3/2)^(1/3)/16 - 89/(18*Pi^(1/3)*zeta(3/2)^(2/3)))/n^(1/3)). - Vaclav Kotesovec, Mar 06 2026

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*k<n or k>n, 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 *)

Crossrefs

Cf. A000217, A007294, A111178, A319797, A393988.

Sequence in context: A319818 A319819 A319820 * A294078 A064133 A295101

Adjacent sequences: A319796 A319797 A319798 * A319800 A319801 A319802

Keyword

nonn

Author

Alois P. Heinz, Sep 28 2018

Status

approved