A359647 a(n) = [x^n] hypergeom([1/4, 3/4], [2], 64*x). The central terms of the Motzkin triangle A359364 without zeros.

1, 6, 140, 4620, 180180, 7759752, 356948592, 17210021400, 859544957700, 44123307828600, 2315270298060720, 123691561681243920, 6707888537328997200, 368417878127146461600, 20455964090297751153600, 1146556787261188952159280, 64797319609481605046295780

Offset

0,2

Comments

Number of Motzkin paths of length 4n with exactly 2n horizontal steps: a(1) = 6: UDHH, UHDH, UHHD, HUDH, HUHD, HHUD. - Alois P. Heinz, Aug 02 2023

Links

Table of n, a(n) for n=0..16.

Formula

a(n) = A359364(4*n, 2*n).

a(n) = A000108(n) * A001448(n) = (binomial(2*n,n)/(n+1))*binomial(4*n,2*n). - Alois P. Heinz, Aug 02 2023

a(n) ~ 2^(6*n-1/2) / (n^2 * Pi). - Amiram Eldar, Oct 11 2025

(2*n + 1) divides a(n) since a(n)/(2*n+1) = Catalan(n)*Catalan(2*n) = A151332(n). - Peter Bala, Jun 27 2026

Maple

ser := series(hypergeom([1/4, 3/4], [2], 64*x), x, 20):
seq(coeff(ser, x, n), n = 0..16);

Mathematica

a[n_] := CatalanNumber[n] * Binomial[4*n, 2*n]; Array[a, 30, 0] (* Amiram Eldar, Oct 11 2025 *)

Prog

(Python)
from math import comb
def a():
    n = 0
    while True:
        yield (comb(2 * n, n) // (n + 1)) * comb(4 * n, 2 * n)
        n += 1
A359647 = a()
for _ in range(17): print(next(A359647))  # Peter Luschny, Jul 01 2026

Crossrefs

Cf. A000108, A001006, A001448, A151332, A359364.

Sequence in context: A193876 A293124 A193185 * A322625 A012785 A012818

Adjacent sequences: A359644 A359645 A359646 * A359648 A359649 A359650

Keyword

nonn,easy,changed

Author

Peter Luschny, Jan 09 2023

Status

approved