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A119309
a(n) = binomial(2*n,n) * 6^n.
3
1, 12, 216, 4320, 90720, 1959552, 43110144, 960740352, 21616657920, 489977579520, 11171488813056, 255928652808192, 5886359014588416, 135839054182809600, 3143703825373593600, 72933928748667371520
OFFSET
0,2
COMMENTS
Number of lattice paths from (0,0) to (n,n) using three kinds of steps (1,0) and two kinds of steps (0,1). - Joerg Arndt, Jul 01 2011
Central terms of the triangles in A013620 and A038220.
LINKS
Hacène Belbachir and Abdelghani Mehdaoui, Recurrence relation associated with the sums of square binomial coefficients, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.
FORMULA
a(n) = 6^n * A000984(n).
G.f.: 1/sqrt(1-24*x). - Zerinvary Lajos, Dec 20 2008 [Corrected by Joerg Arndt, Jul 01 2011]
D-finite with recurrence: n*a(n) +12*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Jan 20 2020
a(n) = 2^n*A098658(n) = 3^n*A059304(n). - R. J. Mathar, Jan 20 2020
From Amiram Eldar, Jul 21 2020: (Start)
Sum_{n>=0} 1/a(n) = 24/23 + 24*sqrt(23)*arcsin(1/sqrt(24))/529.
Sum_{n>=0} (-1)^n/a(n) = 24/25 - 24*arcsinh(1/sqrt(24))/125. (End)
E.g.f.: exp(12*x) * BesselI(0,12*x). - Ilya Gutkovskiy, Sep 14 2021
EXAMPLE
a(3) = binomial(2*3,3) * (6^3) = 20 * 216 = 4320. - Indranil Ghosh, Mar 03 2017
MATHEMATICA
Table[Binomial[2n, n]*(6^n), {n, 0, 15}] (* Indranil Ghosh, Mar 03 2017 *)
PROG
(PARI) /* same as in A092566 but use */
steps=[[1, 0], [1, 0], [1, 0], [0, 1], [0, 1]]; /* note repeated entries */
/* Joerg Arndt, Jun 30 2011 */
(PARI) a(n)=binomial(2*n, n)*6^n \\ Charles R Greathouse IV, Mar 03 2017
(Python)
import math
f=math.factorial
def C(n, r): return f(n)//f(r)//f(n-r)
def A119309(n): return C(2*n, n)*(6**n) # Indranil Ghosh, Mar 03 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, May 14 2006
STATUS
approved