A359067 a(2*n) = Sum_{k=0..n-1} binomial(2*n,k) binomial(2*n-1-k, n-1-k). a(2*n+1) = (Sum_{k=0..n} binomial(2*n+1,k) binomial(2*n-k, n-k)) - binomial(2*n-1, n).

0, 1, 4, 7, 28, 49, 199, 351, 1436, 2561, 10499, 18943, 77617, 141569, 579149, 1066495, 4354780, 8085505, 32954635, 61616127, 250713893, 471556097, 1915928117, 3621830655, 14696701553, 27902803969, 113099318869, 215530668031, 872780984131, 1668644405249, 6751457741849

Offset: 1

Author

Bridget Tenner, Dec 15 2022

Keywords

• easy
• nonn

Comments

For n >= 3, the number of admissible pinnacle sets in the group S_n^D of even-signed permutations.

The even-indexed terms match the even-indexed terms of A359066. The odd-indexed terms differ from the odd-indexed terms of A359066 by binomial(2*n-1, n).

Examples

			For n = 3, the a(3) = 4 admissible pinnacle sets in S_3^D are {}, {1}, {2}, {3}.
		

Links

• Nicolle González, Pamela E. Harris, Gordon Rojas Kirby, Mariana Smit Vega Garcia, and Bridget Eileen Tenner, Pinnacle sets of signed permutations, arXiv:2301.02628 [math.CO] (2023).

Crossrefs

Cf. A289871, A359066, A359068, A178792, A240721.

Programs

Maple

a := n -> if irem(n - 1, 2) = 1 then binomial(n, n/2 - 1)*hypergeom([n/2 + 1, -n/2 + 1], [n/2 + 2], -1) else binomial(n + 1, n/2 + 1/2)*hypergeom([n/2 + 1/2, -n/2 + 1/2], [n/2 + 3/2], -1)/2 - binomial(n - 2, n/2 - 1/2) fi:
seq(simplify(a(n)), n = 3..31); # Peter Luschny, Jan 03 2023

Formula

a(2*n) = Sum_{k=0..n-1} binomial(2*n,k) binomial(2*n-1-k, n-1-k).

a(2*n+1) = (Sum_{k=0..n} binomial(2*n+1,k) binomial(2*n-k, n-k)) - binomial(2*n-1, n).

a(n) = A240721((n-2)/2) if n-1 is odd and otherwise A178792((n-1)/2) - binomial(2*n - 1, n). - Peter Luschny, Jan 03 2023