A357641 Number of integer compositions of 2n whose half-alternating sum is 0.

1, 0, 2, 8, 28, 104, 396, 1504, 5720, 21872, 83980, 323344, 1248072, 4828784, 18721080, 72711552, 282861360, 1101980000, 4298748300, 16789002736, 65641204200, 256895795312, 1006308200040, 3945185586368, 15478849767888, 60774329914144, 238775589937976

Offset

0,3

Comments

We define the half-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A + B - C - D + E + F - G - ...

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1664

Example

The a(0) = 1 through a(3) = 8 compositions:
  ()  .  (112)   (123)
         (1111)  (213)
                 (1212)
                 (1221)
                 (2112)
                 (2121)
                 (11121)
                 (11211)

Maple

a:= proc(n) option remember; `if`(n<3, [1, 0, 2][n+1],
      (8*(n-3)*(5*n-7)*(2*n-5)*a(n-3) -4*(5*n-12)*(n-2)^2*a(n-2)
       +2*(2*n-5)*(5*n-7)*n*a(n-1))/((5*n-12)*(n+1)*(n-2)))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Oct 19 2022

Mathematica

halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]), {i, Length[f]}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[2n], halfats[#]==0&]], {n, 0, 7}]

Crossrefs

The skew-alternating version appears to be A001700.

The version for partitions is A035363.

The skew-alternating form is A088218 (also for full alternating sum).

These compositions are ranked by A357625, reverse A357626.

For reversed partitions we have A357639, ranked by A357631.

A124754 gives alternating sum of standard compositions, reverse A344618.

A357621 = half-alternating sum of standard compositions, reverse A357622.

A357637 counts partitions by half-alternating sum, skew A357638.

Cf. A000583, A001511, A053251, A344619, A357136, A357182, A357627, A357628, A357629, A357633, A357642.

Sequence in context: A114590 A133592 A115967 * A150714 A292668 A122447

Adjacent sequences: A357638 A357639 A357640 * A357642 A357643 A357644

Keyword

nonn

Author

Gus Wiseman, Oct 12 2022

Extensions

a(11)-a(26) from Alois P. Heinz, Oct 19 2022

Status

approved