A340766 Number of ordered subsequences of {1,...,2n} containing at least n elements and such that the first differences contain only odd numbers.

1, 3, 7, 17, 43, 106, 273, 678, 1759, 4389, 11430, 28614, 74685, 187433, 489926, 1231957, 3223387, 8118434, 21256897, 53609282, 140442534, 354595210, 929326086, 2348710733, 6157476873, 15575365846, 40843347873, 103392210473, 271181242774, 686944588009

Offset: 0

Author

Alois P. Heinz, Jun 10 2021

Keywords

• nonn

Examples

			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].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2413

Crossrefs

Cf. A345123.

Programs

Maple

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);

Mathematica

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 *)

Formula

a(n) = A345123(2n,n).

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