A027188 a(n) = number of partitions of n into an odd number of parts, the least being 2; also a(n+2) = number of partitions of n into an even number of parts, each >=2.

0, 0, 1, 0, 0, 0, 1, 1, 2, 2, 4, 4, 6, 7, 11, 12, 17, 20, 28, 33, 44, 52, 69, 82, 105, 126, 161, 191, 239, 286, 355, 423, 520, 618, 755, 896, 1084, 1285, 1549, 1829, 2190, 2583, 3079, 3621, 4297, 5041, 5960, 6977, 8214, 9595, 11264, 13123, 15353, 17854, 20828

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Links

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

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n=0, t, `if`(i<2, 0,
       b(n, i-1, t) +`if`(i>n, 0, b(n-i, i, 1-t))))
    end:
a:= n-> b(n-2$2, 1):
seq(a(n), n=0..80);  # Alois P. Heinz, Feb 27 2014

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[n==0, t, If[i<2, 0, b[n, i-1, t] + If[i>n, 0, b[n-i, i, 1-t]]]]; a[n_] := b[n-2, n-2, 1]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Apr 08 2015, after Alois P. Heinz *)

PARI

gf=sum(n=0, N,  q^(2*n+1)/prod(k=1,n, 1-q^(2*k)) * q^(2*n+1)/prod(k=1,n, 1-q^(2*k-1)) );
concat([0], Vec(gf) ) \\ Joerg Arndt, Feb 27 2014

Formula

G.f.: sum(n>=0, q^(2*n+1)/prod(k=1..n, 1-q^(2*k)) * q^(2*n+1)/prod(k=1..n, 1-q^(2*k-1)) ). [Joerg Arndt, Feb 27 2014]

a(n+2) + A027194(n+2) = A002865(n). - R. J. Mathar, Jun 16 2018

G.f.: x^2 * Sum_{k>=0} x^(4*k)/Product_{j=1..2*k} (1-x^j). - Seiichi Manyama, May 15 2023

a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3*2^(7/2)*n^(3/2)). - Vaclav Kotesovec, Jun 20 2025