cp's OEIS Frontend

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

%I A357641 #15 Oct 19 2022 19:04:07
%S A357641 1,0,2,8,28,104,396,1504,5720,21872,83980,323344,1248072,4828784,
%T A357641 18721080,72711552,282861360,1101980000,4298748300,16789002736,
%U A357641 65641204200,256895795312,1006308200040,3945185586368,15478849767888,60774329914144,238775589937976
%N A357641 Number of integer compositions of 2n whose half-alternating sum is 0.
%C A357641 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 - ...
%H A357641 Alois P. Heinz, <a href="/A357641/b357641.txt">Table of n, a(n) for n = 0..1664</a>
%e A357641 The a(0) = 1 through a(3) = 8 compositions:
%e A357641   ()  .  (112)   (123)
%e A357641          (1111)  (213)
%e A357641                  (1212)
%e A357641                  (1221)
%e A357641                  (2112)
%e A357641                  (2121)
%e A357641                  (11121)
%e A357641                  (11211)
%p A357641 a:= proc(n) option remember; `if`(n<3, [1, 0, 2][n+1],
%p A357641       (8*(n-3)*(5*n-7)*(2*n-5)*a(n-3) -4*(5*n-12)*(n-2)^2*a(n-2)
%p A357641        +2*(2*n-5)*(5*n-7)*n*a(n-1))/((5*n-12)*(n+1)*(n-2)))
%p A357641     end:
%p A357641 seq(a(n), n=0..40);  # _Alois P. Heinz_, Oct 19 2022
%t A357641 halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]),{i,Length[f]}];
%t A357641 Table[Length[Select[Join@@Permutations/@IntegerPartitions[2n],halfats[#]==0&]],{n,0,7}]
%Y A357641 The skew-alternating version appears to be A001700.
%Y A357641 The version for partitions is A035363.
%Y A357641 The skew-alternating form is A088218 (also for full alternating sum).
%Y A357641 These compositions are ranked by A357625, reverse A357626.
%Y A357641 For reversed partitions we have A357639, ranked by A357631.
%Y A357641 A124754 gives alternating sum of standard compositions, reverse A344618.
%Y A357641 A357621 = half-alternating sum of standard compositions, reverse A357622.
%Y A357641 A357637 counts partitions by half-alternating sum, skew A357638.
%Y A357641 Cf. A000583, A001511, A053251, A344619, A357136, A357182, A357627, A357628, A357629, A357633, A357642.
%K A357641 nonn
%O A357641 0,3
%A A357641 _Gus Wiseman_, Oct 12 2022
%E A357641 a(11)-a(26) from _Alois P. Heinz_, Oct 19 2022