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

0,2

Links

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

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

Example

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].

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 *)

Crossrefs

Cf. A345123.

Sequence in context: A178778 A335596 A238824 * A161943 A134184 A142975

Adjacent sequences: A340763 A340764 A340765 * A340767 A340768 A340769

Keyword

nonn

Author

Alois P. Heinz, Jun 10 2021

Status

approved