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

0,3

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

Links

Table of n, a(n) for n=0..33.

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)

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 *)

Prog

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

Crossrefs

Cf. A097893, A128017, A128018.

Cf. A000984, A249095, A249308.

Sequence in context: A309094 A109859 A128057 * A135401 A129881 A132369

Adjacent sequences: A128011 A128012 A128013 * A128015 A128016 A128017

Keyword

easy,nonn

Author

Paul Barry, Feb 11 2007

Status

approved