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

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Index entries for sequences related to compositions

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).

Example

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

Maple

b:= proc(n, i, p) option remember; `if`(i*(i+1)/2<n, 0, `if`(n=0,
      irem(p+1, 2)*p!, add(b(n-i*j, i-1, p+j), j=0..min(1, n/i))))
    end:
a:= n-> 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]

Crossrefs

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

Sequence in context: A259881 A238132 A278296 * A340282 A008642 A001364

Adjacent sequences: A332302 A332303 A332304 * A332306 A332307 A332308

Keyword

nonn

Author

Ilya Gutkovskiy, Feb 09 2020

Status

approved