A027195 Number of partitions of n into an even number of parts, the least being 3; also, a(n+3) = number of partitions of n into an odd number of parts, each >=3.

0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 9, 10, 13, 16, 20, 24, 31, 36, 45, 54, 66, 78, 97, 114, 138, 164, 197, 232, 280, 328, 392, 461, 546, 639, 758, 884, 1041, 1215, 1425, 1657, 1941, 2250, 2624, 3041, 3534, 4084, 4740, 5465, 6321, 7280, 8399

Offset

1,12

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Formula

a(n) + A027189(n) = A026796(n). - R. J. Mathar, Oct 18 2019

a(n) ~ Pi^2 * exp(Pi*sqrt(2*n/3)) / (8 * 3^(3/2) * n^2). - Vaclav Kotesovec, May 17 2020

G.f.: x^6 * Sum_{k>=0} x^(6*k)/Product_{j=1..2*k+1} (1-x^j). - Seiichi Manyama, May 15 2023

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<3, 0, b(n-3, 3, 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 < 3, 0, b[n - 3, 3, 0]];
Array[a, 100] (* Jean-François Alcover, May 17 2020, after Alois P. Heinz *)

Crossrefs

Cf. A026796, A027189.

Sequence in context: A245439 A132326 A351595 * A008483 A281356 A026796

Adjacent sequences: A027192 A027193 A027194 * A027196 A027197 A027198

Keyword

nonn

Author

Clark Kimberling

Status

approved