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g[n_, k_] := g[n, k] = Which[k > n, 0, k == 0, 1, k == 1, n,

True, g[n - 1, k - 1] + g[n - 2, k]];

b[n_, k_] := b[n, k] = If[k > n, 0, g[n, k] + b[n, k + 1]];

a[n_] := b[2*n, n];

Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 29 2022, after Alois P. Heinz *)

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a(n) ~ c * (27/4)^(n/2) / sqrt(3*Pi*n/2), where c = 14 if n is even and c = 8*sqrt(3) if n is odd. Equivalently, c = 7 + 4*sqrt(3) + (7 - 4*sqrt(3))*(-1)^n. - Vaclav Kotesovec, Jun 19 2021

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Alois P. Heinz, <a href="/A340766/b340766.txt">Table of n, a(n) for n = 0..2413</a>

g:= proc(n, k) option remember; `if`(k>n, 0,

`if`(k in [0, 1], n^k, g(n-1, k-1)+g(n-2, k)))

end:

b:= proc(n, k) option remember;

`if`(k>n, 0, g(n, k)+b(n, k+1))

end:

a:= n-> b(2*n, n):

seq(a(n), n=0..30);

a(3) = 17: [1,2,3], [1,2,5], [1,4,5], [2,3,4], [2,3,6], [2,5,6], [3,4,5], [4,5,6], [1,2,3,4], [1,2,3,6], [1,2,5,6], [1,4,5,6], [2,3,4,5], [3,4,5,6], [1,2,3,4,5], [2,3,4,5,6], [1,2,3,4,5,6].

a

Number of ordered subsequences of {1,...,2n} containing at least n elements and such that the first differences contain only odd numbers.