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

0,4

Links

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

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

Example

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.

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 *)

Prog

(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]

Crossrefs

Cf. A007294.

Sequence in context: A330952 A118399 A278298 * A076571 A084783 A265853

Adjacent sequences: A178924 A178925 A178926 * A178928 A178929 A178930

Keyword

nonn,easy

Author

David S. Newman, Dec 29 2010

Status

approved