A178927 Number of partitions into a triangular number of parts.

1, 1, 1, 2, 2, 3, 5, 6, 8, 11, 15, 19, 26, 32, 42, 54, 69, 86, 111, 137, 173, 215, 268, 329, 409, 499, 614, 748, 914, 1106, 1346, 1621, 1958, 2352, 2827, 3380, 4048, 4821, 5746, 6824, 8102, 9587, 11346, 13383, 15781, 18566, 21824, 25597, 30007, 35100, 41029

Offset: 0

Author

David S. Newman, Dec 29 2010

Keywords

• nonn
• easy

Examples

			For example there are 7 unrestricted partitions of 5, namely: 5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1 and 1+1+1+1+1. Of these we allow only those with 1,3,6,10,... parts.  These are 5, 3+1+1 and 2+2+1. So a(5)=3.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..8000 (terms 0..2000 from Alois P. Heinz)

Crossrefs

Cf. A007294.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n=0 or i=1, `if`(issqr(
      1+8*(t+n)), 1, 0), b(n, i-1, t)+b(n-i, min(i, n-i), t+1))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, Jul 29 2017

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[n == 0 || i == 1, If[Sqrt[1 + 8(t + n)] // IntegerQ, 1, 0], b[n, i - 1, t] + b[n - i, Min[i, n - i], t + 1]];
a[n_] := b[n, n, 0];
a /@ Range[0, 80] (* Jean-François Alcover, Nov 11 2020, after Alois P. Heinz *)

Sage

A178927 = lambda n: 1 if n == 0 else sum(number_of_partitions(n,k=tri) for tri in [1..n] if is_triangular_number(tri)) # [D. S. McNeil, Dec 30 2010]

Formula

G.f.: Sum_{i>=0} x^(i*(i+1)/2) / Product_{j=1..i*(i+1)/2} (1 - x^j). - Ilya Gutkovskiy, May 07 2017