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

Author

Peter Luschny, Jan 09 2023

Keywords

• nonn

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

Crossrefs

Cf. A000108, A001006, A001448, A359364.

Programs

Maple

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