A027196 Number of partitions of n into an even number of parts, the least being 4; also, a(n+4) = number of partitions of n into an odd number of parts, each >=4.

0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 12, 14, 17, 20, 25, 29, 35, 41, 50, 58, 70, 81, 97, 113, 134, 156, 185, 214, 252, 292, 343, 396, 463, 534, 623, 718, 833, 958, 1110, 1274, 1471, 1686, 1943, 2223, 2555, 2919, 3347, 3818, 4368

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Crossrefs

Cf. A026797, A027190.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n=0, t,
     `if`(i>n, 0, b(n, i+1, t)+b(n-i, i, 1-t)))
    end:
a:= n-> `if`(n<4, 0, b(n-4, 4, 0)):
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 18 2019

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[n == 0, t, If[i > n, 0, b[n, i + 1, t] + b[n - i, i, 1 - t]]];
a[n_] := If[n < 4, 0, b[n - 4, 4, 0]];
Array[a, 100] (* Jean-François Alcover, May 17 2020, after Alois P. Heinz *)

Formula

a(n) + A027190(n) = A026797(n). - R. J. Mathar, Oct 18 2019

G.f.: x^8 * Sum_{k>=0} x^(8*k)/Product_{j=1..2*k+1} (1-x^j). - Seiichi Manyama, May 15 2023