A059304 a(n) = 2^n * (2*n)! / (n!)^2.

1, 4, 24, 160, 1120, 8064, 59136, 439296, 3294720, 24893440, 189190144, 1444724736, 11076222976, 85201715200, 657270374400, 5082890895360, 39392404439040, 305870434467840, 2378992268083200, 18531097667174400

Offset

0,2

Comments

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

The convolution square root of this sequence is A004981. - T. D. Noe, Jun 11 2002

Also main diagonal of array: T(i,1)=2^(i-1), T(1,j)=1, T(i,j) = T(i,j-1) + 2*T(i-1,j). - Benoit Cloitre, Feb 26 2003

The Hankel transform (see A001906 for definition) of this sequence with interpolated zeros(1, 0, 4, 0, 24, 0, 160, 0, 1120, ...) = is A036442: 1, 4, 32, 512, 16384, ... . - Philippe Deléham, Jul 03 2005

The Hankel transform of this sequence gives A103488. - Philippe Deléham, Dec 02 2007

Equals the central column of the triangle A038207. - Zerinvary Lajos, Dec 08 2007

Links

Harry J. Smith, Table of n, a(n) for n = 0..200

Paul Barry and Arnauld Mesinga Mwafise, Classical and Semi-Classical Orthogonal Polynomials Defined by Riordan Arrays, and Their Moment Sequences, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.5.

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.

H. J. Brothers, Pascal's Prism: Supplementary Material.

Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

Formula

a(n) = C(2*n,n) * 2^n.

D-finite with recurrence a(n) = a(n-1)*(8-4/n).

a(n) = A000079(n)*A000984(n)

G.f.: 1/sqrt(1-8*x) - T. D. Noe, Jun 11 2002

E.g.f.: exp(4*x)*BesselI(0, 4*x). - Vladeta Jovovic, Aug 20 2003

a(n) = A038207(n,n). - Joerg Arndt, Jul 01 2011

G.f.: G(0)/2, where G(k) = 1 + 1/(1 - 4*x*(2*k+1)/(4*x*(2*k+1) + (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 24 2013

E.g.f.: E(0)/2, where E(k) = 1 + 1/(1 - 4*x/(4*x + (k+1)^2/(2*k+1)/E(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 01 2013

G.f.: Q(0)/(1+2*sqrt(x)), where Q(k) = 1 + 2*sqrt(x)/(1 - 2*sqrt(x)*(2*k+1)/(2*sqrt(x)*(2*k+1) + (k+1)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 09 2013

O.g.f.: hypergeom([1/2], [], 8*x). - Peter Luschny, Oct 08 2015

a(n) = Sum_{k = 0..2*n} (-1)^(n+k)*binomial(2*n,k)*binomial(3*n - 2*k,n)*binomial(n+k,n). - Peter Bala, Aug 04 2016

a(n) ~ 8^n/sqrt(Pi*n). - Ilya Gutkovskiy, Aug 04 2016

From Amiram Eldar, Jul 21 2020: (Start)

Sum_{n>=0} 1/a(n) = 8/7 + 8*sqrt(7)*arcsin(1/sqrt(8))/49.

Sum_{n>=0} (-1)^n/a(n) = (8/27)*(3 - arcsinh(1/sqrt(8))). (End)

a(n) = Sum_{k = n..2*n} binomial(2*n,k)*binomial(k,n). In general, for m >= 1, Sum_{k = n..m*n} binomial(m*n,k)*binomial(k,n) = 2^((m-1)*n)*binomial(m*n,n). - Peter Bala, Mar 25 2023

Maple

seq(binomial(2*n, n)*2^n, n=0..19); # Zerinvary Lajos, Dec 08 2007

Mathematica

Table[2^n Binomial[2n, n], {n, 0, 30}] (* Harvey P. Dale, Dec 16 2014 *)

Prog

(PARI) {a(n)=if(n<0, 0, 2^n*(2*n)!/n!^2)} /* Michael Somos, Jan 31 2007 */
(PARI) { for (n = 0, 200, write("b059304.txt", n, " ", 2^n * (2*n)! / n!^2); ) } \\ Harry J. Smith, Jun 25 2009
(PARI) /* as lattice paths: same as in A092566 but use */
steps=[[1, 0], [1, 0], [0, 1]]; /* note the double [1, 0] */
/* Joerg Arndt, Jul 01 2011 */
(Magma) [2^n*Factorial(2*n)/Factorial(n)^2: n in [0..25]]; // Vincenzo Librandi, Oct 08 2015

Crossrefs

Diagonal of A013609.

Cf. A038207.

Column k=0 of A067001.

Sequence in context: A117337 A272865 A084130 * A069722 A343842 A027079

Adjacent sequences: A059301 A059302 A059303 * A059305 A059306 A059307

Keyword

nonn,easy

Author

Henry Bottomley, Jan 25 2001

Status

approved