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

0,8

Links

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

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

Example

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.

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 *)

Prog

(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
(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

Crossrefs

Cf. A000009, A024940, A225044, A087154.

Sequence in context: A291323 A290988 A165025 * A361736 A379314 A278575

Adjacent sequences: A225042 A225043 A225044 * A225046 A225047 A225048

Keyword

nonn

Author

Reinhard Zumkeller, Apr 25 2013

Status

approved