A350265 a(n) = hypergeometric([-n - 1, 1 - n, -n], [1, 3], -1).

1, 1, 3, 12, 55, 276, 1477, 8296, 48393, 291010, 1794320, 11297760, 72413640, 471309944, 3108745785, 20746732688, 139899430981, 952127880138, 6533934575018, 45175430719240, 314467004704818, 2202576030828096, 15514620388706488, 109851319423632192, 781531332298053400

Offset: 0

Author

Peter Luschny, Dec 28 2021

Keywords

• nonn

Links

• T. Amdeberhan, A rather curious identity on sums over triple binomial terms, MathOverflow 2021.

Crossrefs

Cf. A000217, A002378, A277188, A001181 (Baxter permutations).

Programs

Maple

a := proc(n) option remember; if n < 2 then 1 else ((n + 1)*((7*n^2 + 7*n - 2)*a(n - 1) + 8*(n - 2)*n*a(n - 2)))/(n*(n + 2)*(n + 3)) fi end:
seq(a(n), n = 0..24);

Mathematica

a[n_] := HypergeometricPFQ[{-n - 1, 1 - n, -n}, {1, 3}, -1];
Table[a[n], {n, 0, 24}]

Python

from sympy import hyperexpand
from sympy.functions import hyper
def A350265(n): return hyperexpand(hyper((-n-1,1-n,-n),(1,3),-1)) # Chai Wah Wu, Dec 29 2021

Formula

a(n) * A000217(n) = Sum_{k=0..n-1} binomial(n + 1, k) * binomial(n, k) * binomial(n + 1, k + 2).

a(n) * A002378(n) = Sum_{k=0..n-1} binomial(n + 1, k) * binomial(n + 1, k + 1) * binomial(n + 1, k + 2).

For a recurrence see the Maple program.

a(n) ~ 2^(3*n+4) / (Pi*sqrt(3)*n^3). - Vaclav Kotesovec, Apr 27 2024