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A128014 Central binomial coefficients C(2n,n) repeated. 13
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 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A309094 A109859 A128057 * A135401 A129881 A132369
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Feb 11 2007
STATUS
approved

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Last modified February 20 20:53 EST 2024. Contains 370217 sequences. (Running on oeis4.)