A119309 - OEIS

%I #31 Sep 15 2021 07:28:05

%S 1,12,216,4320,90720,1959552,43110144,960740352,21616657920,

%T 489977579520,11171488813056,255928652808192,5886359014588416,

%U 135839054182809600,3143703825373593600,72933928748667371520

%N a(n) = binomial(2*n,n) * 6^n.

%C 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

%C Central terms of the triangles in A013620 and A038220.

%H Indranil Ghosh, <a href="/A119309/b119309.txt">Table of n, a(n) for n = 0..400</a>

%H Hacène Belbachir and Abdelghani Mehdaoui, <a href="https://doi.org/10.2989/16073606.2020.1729269">Recurrence relation associated with the sums of square binomial coefficients</a>, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.

%F a(n) = 6^n * A000984(n).

%F G.f.: 1/sqrt(1-24*x). - _Zerinvary Lajos_, Dec 20 2008 [Corrected by _Joerg Arndt_, Jul 01 2011]

%F D-finite with recurrence: n*a(n) +12*(-2*n+1)*a(n-1)=0. - _R. J. Mathar_, Jan 20 2020

%F a(n) = 2^n*A098658(n) = 3^n*A059304(n). - _R. J. Mathar_, Jan 20 2020

%F From _Amiram Eldar_, Jul 21 2020: (Start)

%F Sum_{n>=0} 1/a(n) = 24/23 + 24*sqrt(23)*arcsin(1/sqrt(24))/529.

%F Sum_{n>=0} (-1)^n/a(n) = 24/25 - 24*arcsinh(1/sqrt(24))/125. (End)

%F E.g.f.: exp(12*x) * BesselI(0,12*x). - _Ilya Gutkovskiy_, Sep 14 2021

%e a(3) = binomial(2*3,3) * (6^3) = 20 * 216 = 4320. - _Indranil Ghosh_, Mar 03 2017

%t Table[Binomial[2n,n]*(6^n), {n, 0, 15}] (* _Indranil Ghosh_, Mar 03 2017 *)

%o (PARI) /* same as in A092566 but use */

%o steps=[[1,0], [1,0], [1,0], [0,1], [0,1]]; /* note repeated entries */

%o /* _Joerg Arndt_, Jun 30 2011 */

%o (PARI) a(n)=binomial(2*n,n)*6^n \\ _Charles R Greathouse IV_, Mar 03 2017

%o (Python)

%o import math

%o f=math.factorial

%o def C(n,r): return f(n)//f(r)//f(n-r)

%o def A119309(n): return C(2*n,n)*(6**n) # _Indranil Ghosh_, Mar 03 2017

%Y Cf. A000984, A013620, A038220, A059304, A098658.

%K nonn

%O 0,2

%A _Reinhard Zumkeller_, May 14 2006