A098658 a(n) = 3^n*(2*n)!/(n!)^2.

1, 6, 54, 540, 5670, 61236, 673596, 7505784, 84440070, 956987460, 10909657044, 124965162504, 1437099368796, 16581915793800, 191876454185400, 2225766868550640, 25874539846901190, 301362287628613860

Offset: 0

Author

Paul Barry, Sep 20 2004

Keywords

• easy
• nonn

Comments

Number of lattice paths from (0,0) to (n,n) using steps (0,1) and three kinds of steps (1,0). - Joerg Arndt, Jul 01 2011

Sixth binomial transform of 1/sqrt(1-36*x^2).

Diagonal of the rational function 1 / (1 - 3*x - y). - Ilya Gutkovskiy, Apr 24 2025

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Hacène Belbachir and Abdelghani Mehdaoui, Recurrence relation associated with the sums of square binomial coefficients, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.
• Hacène Belbachir, Abdelghani Mehdaoui, and László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
• Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Crossrefs

Cf. A000984, A059304, A101600, A322246.

Programs

Magma

[3^n*Factorial(2*n)/Factorial(n)^2: n in [0..20]]; // Vincenzo Librandi, Jul 05 2011

Mathematica

Table[3^n (2n)!/(n!)^2,{n,0,20}] (* Harvey P. Dale, Dec 14 2011 *)

PARI

/* same as in A092566 but use */
steps=[[1,0], [1,0], [1,0], [0,1]]; /* note the triple [1,0] */
/* Joerg Arndt, Jun 30 2011 */

Formula

G.f.: 1/sqrt((1-6*x)^2-36*x^2) = 1/sqrt(1-12*x).

E.g.f.: exp(6*x)*BesselI(0, 6x).

a(n) = [t^n](1+6*t+9*t^2)^n.

a(n) = 3^n*A000984(n). - R. J. Mathar, Oct 10 2012

G.f.: Q(0), where Q(k) = 1 + 12*x*(4*k+1)/( 4*k+2 - 12*x*(4*k+2)*(4*k+3)/(12*x*(4*k+3) + 4*(k+1)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 14 2013

n*a(n) +6*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Nov 27 2014

exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 6*x + 45*x^2 + 378*x^3 + ... is the o.g.f. for A101600. - Peter Bala, Jul 16 2015

From Amiram Eldar, Jul 21 2020: (Start)

Sum_{n>=0} 1/a(n) = 12/11 + 12*sqrt(11)*arcsin(1/sqrt(12))/121.

Sum_{n>=0} (-1)^n/a(n) = 12/13 - 12*sqrt(13)*arcsinh(1/sqrt(12))/169. (End)

From Peter Bala, Oct 12 2024: (Start)

a(n) = Integral_{x = 0..12} x^n * w(x) dx, where w(x) = 1/( Pi*sqrt(x*(12 - x)) ) is positive on the interval (0, 12). The weight function w(x) is singular at x = 0 and at x = 12 and is the solution of the Hausdorff moment problem.

Binomial transform of A322246.

The Gauss congruences hold: a(n*p^r) == a(n*p^(r-1)) (mod p^r) for all primes p and positive integers n and r. (End)