A098430 a(n) = 4^n*(2*n)!/(n!)^2.

1, 8, 96, 1280, 17920, 258048, 3784704, 56229888, 843448320, 12745441280, 193730707456, 2958796259328, 45368209309696, 697972450918400, 10768717814169600, 166556168859156480, 2581620617316925440, 40091049586568724480, 623638549124402380800, 9715632133727531827200

Offset: 0

Author

Paul Barry, Sep 07 2004

Keywords

• easy
• nonn
• changed

Comments

a(n) counts walks of 2n steps North, East, South or West that start at the origin and end on the line y=x. For example, a(1)=8 counts EW, EN, NE, NS, WE, WS, SN, SW. If the walk has i East and j North steps, then it must have n-j West and n-i South steps. There are Multinomial[i,j,n-j,n-i] ways to arrange these steps and summing over i and j gives the result. - David Callan, Oct 11 2005

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

Hankel transform is A121913. - Philippe Deléham, Mar 01 2009

Convolving a(n) with itself yields A001025, the powers of 16. Thus the limiting ratio of this sequence is 16. - Bob Selcoe, Jul 16 2014

Number of strings x of length 4n over the alphabet {1, -1} such that the dot product of x with (x reversed) is 0. - Jeffrey Shallit, Mar 06 2017

Number of orthogonal pairs of vectors of length 2n, constructed with any symmetric binary-valued symbol set. - Ross Drewe, May 18 2018

Diagonal of the rational function 1 / (1 - 4*x - y). - Ilya Gutkovskiy, Apr 24 2025

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• 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. A000302, A000984, A174301, A249308.

Programs

Haskell

a098430 n = a000302 n * a000984 n -- Reinhard Zumkeller, Nov 14 2014

Magma

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

Maple

A098430 := n -> 4^n*binomial(2*n,n): seq(A098430(n), n=0..30); # Wesley Ivan Hurt, Jul 16 2014

Mathematica

CoefficientList[Series[1/Sqrt[1 - 16 x], {x, 0, 16}], x] (* Robert G. Wilson v, Jun 28 2012 *)
Table[4^n(2n)!/(n!)^2,{n,0,20}] (* Harvey P. Dale, Aug 13 2021 *)

PARI

/* as lattice paths: same as in A092566 but use */
steps=[[1,0], [1,0], [0,1], [0,1]]; /* note the double [1,0] and [0,1] */
/* Joerg Arndt, Jul 01 2011 */

Sage

a = lambda n: 16^n*hypergeometric([-2*n, 1/2], [1], 2)
[simplify(a(n)) for n in range(23)] # Peter Luschny, May 19 2015

Formula

a(n) = 4^n*binomial(2*n, n) = 4^n*A000984(n).

E.g.f.: exp(8*x)*BesselI(0, 8*x).

G.f.: 1/sqrt(1-16*x). - Zerinvary Lajos, Dec 20 2008, corrected R. J. Mathar, May 18 2009

a(n) = (1/Pi)*Integral_{x=-2..2} (2*x)^(2*n)/sqrt((2-x)*(2+x)) dx. - Peter Luschny, Sep 12 2011

D-finite with recurrence: n*a(n) + 8*(-2*n+1)*a(n-1) = 0. - R. J. Mathar, Nov 10 2014

a(n) = A249308(2*n). - Reinhard Zumkeller, Nov 14 2014

a(n) = 16^n*hypergeometric([-2*n, 1/2], [1], 2). - Peter Luschny, May 19 2015

a(n) = A174301(2n,n). - Alois P. Heinz, Apr 15 2019

From Amiram Eldar, Jul 21 2020: (Start)

Sum_{n>=0} 1/a(n) = 16/15 + 16*sqrt(15)*arcsin(1/4)/225.

Sum_{n>=0} (-1)^n/a(n) = 16/17 - 16*sqrt(17)*arcsinh(1/4)/289. (End)

a(n) = Sum_{k = 0..2*n} (-1)^k *A000984(k) * A000984(2*n-k). Cf. Sum_{k = 0..2*n} A000984(k) * A000984(2*n-k) = 16^n. - Peter Bala, Aug 23 2025