A128014 Central binomial coefficients C(2n,n) repeated.

1, 1, 2, 2, 6, 6, 20, 20, 70, 70, 252, 252, 924, 924, 3432, 3432, 12870, 12870, 48620, 48620, 184756, 184756, 705432, 705432, 2704156, 2704156, 10400600, 10400600, 40116600, 40116600, 155117520, 155117520, 601080390, 601080390

Offset: 0

Author

Paul Barry, Feb 11 2007

Keywords

• easy
• nonn

Comments

Binomial transform is A097893. Hankel transform is A128017.

Hankel transform of a(n+1) is A128018. - Paul Barry, Nov 23 2009

Number of 2n-bead balanced binary necklaces that are equivalent to their reverse. - Andrew Howroyd, Sep 29 2017

Number of ballot sequences of length n in which the vote is tied or decided by 1 vote. - Nachum Dershowitz, Aug 12 2020

Number of binary strings of length n that are abelian squares. - Michael S. Branicky, Dec 21 2020

Crossrefs

Cf. A097893, A128017, A128018.

Cf. A000984, A249095, A249308.

Programs

Haskell

a128014 = a000984 . flip div 2
-- Reinhard Zumkeller, Nov 14 2014

Mathematica

(1+x)/Sqrt[1-4x^2] + O[x]^34 // CoefficientList[#, x]& (* Jean-François Alcover, Oct 07 2017 *)
With[{cb=Table[Binomial[2n,n],{n,0,20}]},Riffle[cb,cb]] (* Harvey P. Dale, Feb 17 2020 *)

Formula

G.f.: (1+x)/sqrt(1-4*x^2).

a(n) = C(n,n/2)*(1+(-1)^n)/2 + C(n-1,(n-1)/2)*(1-(-1)^n)/2.

a(n) = (1/Pi)*Integral_{x=-2..2} x^n*(1+x)/(x*sqrt(4-x^2)), as moment sequence.

E.g.f. of a(n+1): Bessel_I(0,2*x)+2*Bessel_I(1,2*x). - Paul Barry, Mar 26 2010

n*a(n) +(n-2)*a(n-1) +4*(-n+1)*a(n-2) +4*(-n+3)*a(n-3) = 0. - R. J. Mathar, Nov 26 2012

a(n) = 2^n*Product_{k=0..n-1} ((k/n+1/n)/2)^((-1)^k). - Peter Luschny, Dec 03 2013

From Reinhard Zumkeller, Nov 14 2014: (Start)

a(n) = A000984(floor(n/2)).

a(n) = A249095(n,n) = A249308(n) / 2^n. (End)