login
A359066
a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n,k)*binomial(n-1-k,floor((n-1)/2) - k).
4
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
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
KEYWORD
easy,nonn
AUTHOR
Bridget Tenner, Dec 15 2022
STATUS
approved