A359066 a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n,k)*binomial(n-1-k,floor((n-1)/2) - k).

1, 1, 5, 7, 31, 49, 209, 351, 1471, 2561, 10625, 18943, 78079, 141569, 580865, 1066495, 4361215, 8085505, 32978945, 61616127, 250806271, 471556097, 1916280833, 3621830655, 14698053631, 27902803969, 113104519169, 215530668031, 872801042431, 1668644405249, 6751535300609

Offset

1,3

Comments

For n >= 3, this is the number of admissible pinnacle sets in the group S_n^B of signed permutations.

The even-indexed terms appear in A240721 and the odd-indexed terms appear in A178792.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1000

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).

Formula

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

a(n) = binomial(n-1, floor((n-1)/2))*hypergeom([-n, ceil((1 -n)/2)], [1 - n], -1). - Peter Luschny, Jan 03 2023

Example

For n = 3, the a(3) = 5 admissible pinnacle sets in S_3^B are {}, {-1}, {1}, {2}, {3}.

Maple

a := n -> add(binomial(n, k)*binomial(n-1-k, iquo(n-1, 2) - k), k = 0..iquo(n-1, 2)):
# Alternative:
a := n -> binomial(n-1, floor((n-1)/2))*hypergeom([-n, ceil((1-n)/2)], [1-n], -1);
seq(simplify(a(n)), n=3..31); # Peter Luschny, Jan 03 2023

Mathematica

Array[Sum[Binomial[#, k]*Binomial[# - 1 - k, Floor[(# - 1)/2] - k], {k, 0, Floor[(# - 1)/2]}] &, 31] (* Michael De Vlieger, Jan 03 2023 *)

Prog

(PARI) a(n) = sum(k=0, (n-1)\2, binomial(n, k)*binomial(n-1-k, (n-1)\2 - k)) \\ Andrew Howroyd, Jan 02 2023

Crossrefs

Cf. A289871, A359067, A359068, A240721, A178792.

Sequence in context: A025095 A025114 A090935 * A153414 A284405 A353285

Adjacent sequences: A359063 A359064 A359065 * A359067 A359068 A359069

Keyword

easy,nonn

Author

Bridget Tenner, Dec 15 2022

Status

approved