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A059304 a(n) = 2^n * (2*n)! / (n!)^2. 18
1, 4, 24, 160, 1120, 8064, 59136, 439296, 3294720, 24893440, 189190144, 1444724736, 11076222976, 85201715200, 657270374400, 5082890895360, 39392404439040, 305870434467840, 2378992268083200, 18531097667174400 (list; graph; refs; listen; history; text; internal format)
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, 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.

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.

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

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: A272865 A084130 * A069722 A027079 A213441 A238299

Adjacent sequences:  A059301 A059302 A059303 * A059305 A059306 A059307

KEYWORD

nonn

AUTHOR

Henry Bottomley, Jan 25 2001

STATUS

approved

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Last modified May 26 21:55 EDT 2018. Contains 304645 sequences. (Running on oeis4.)