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

Author

Gus Wiseman, Oct 12 2022

Keywords

• nonn

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 - ...

Examples

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

Links

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

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.

Programs

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}]

Extensions

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