A332305 Number of compositions (ordered partitions) of n into distinct parts such that number of parts is even.

1, 0, 0, 2, 2, 4, 4, 6, 6, 8, 32, 34, 58, 84, 132, 158, 230, 280, 376, 450, 570, 1388, 1556, 2398, 3310, 4920, 6600, 9674, 12122, 16684, 21340, 28110, 34974, 45392, 55208, 69274, 124498, 143676, 204012, 270758, 377966, 493024, 690304, 895434, 1223826, 1562948

Offset: 0

Author

Ilya Gutkovskiy, Feb 09 2020

Keywords

• nonn

Examples

			a(5) = 4 because we have [4, 1], [3, 2], [2, 3] and [1, 4].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..5000
• Index entries for sequences related to compositions

Crossrefs

Cf. A027187, A032020, A034008, A067661, A332304.

Programs

Maple

b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 b(n$2, 0):
seq(a(n), n=0..55);  # Alois P. Heinz, Feb 09 2020

Mathematica

nmax = 45; CoefficientList[Series[Sum[(2 k)! x^(k (2 k + 1))/Product[1 - x^j, {j, 1, 2 k}], {k, 0, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Sum_{k>=0} (2*k)! * x^(k*(2*k + 1)) / Product_{j=1..2*k} (1 - x^j).

a(n) = A032020(n) - A332304(n).