A225045 Number of partitions of n into distinct non-triangular numbers, cf. A014132.

1, 0, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 3, 5, 5, 5, 7, 7, 10, 10, 13, 13, 16, 18, 21, 25, 27, 32, 33, 41, 44, 53, 57, 65, 73, 81, 93, 102, 118, 128, 145, 159, 181, 200, 224, 246, 275, 304, 337, 375, 413, 460, 503, 559, 614, 679, 749, 821, 907, 991, 1096, 1197, 1319, 1442, 1582, 1733, 1893, 2076, 2265, 2482, 2702, 2956, 3220

Offset: 0

Author

Reinhard Zumkeller, Apr 25 2013

Keywords

• nonn

Examples

			a(10) = #{8+2} = 1;
a(11) = #{11, 9+2, 7+4, 5+4+2} = 4;
a(12) = #{12, 8+4, 7+5} = 3;
a(13) = #{13, 11+2, 9+4, 8+5, 7+4+2} = 5.
		

Links

• Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Crossrefs

Cf. A000009, A024940, A225044, A087154.

Programs

Haskell

a225045 = p a014132_list where
   p _      0 = 1
   p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m

Maple

b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0,
      `if`(n=0, 1, add(b(n-i*j, i-1), j=0..min(n/i,
      `if`(issqr(8*i+1), 0, 1)))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..80);  # Alois P. Heinz, Apr 01 2014

Mathematica

b[n_, i_] := b[n, i] = If[n > i*(i+1)/2, 0, If[n==0, 1, Sum[b[n-i*j, i-1], {j, 0, Min[n/i, If[IntegerQ[Sqrt[8*i+1]], 0, 1]]}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jan 15 2016, after Alois P. Heinz *)

PARI

N=66; q='q+O('q^N); Vec( prod(n=1,N, 1 + q^n) / prod(n=1,N, 1 + q^(n*(n+1)/2)) ) \\ Joerg Arndt, Apr 01 2014

Formula

G.f.: prod(n>=1, 1 + q^n ) / prod(n>=1, 1 + q^(n*(n+1)/2) ). [Joerg Arndt, Apr 01 2014]

a(n) ~ exp(Pi*sqrt(n/3) - 3^(1/4) * Zeta(3/2) * n^(1/4) / (2+sqrt(2)) - 3*(3-2*sqrt(2)) * Zeta(3/2)^2 / (16*Pi)) / (2*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Jan 02 2017