A240721 -id:A240721 - cp's OEIS frontend

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

Bridget Tenner, Dec 15 2022

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.

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

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).

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

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

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

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

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

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

Original entry on oeis.org

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

Bridget Tenner, Dec 15 2022

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).

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

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).

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

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

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

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

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