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A128014
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Central binomial coefficients C(2n,n) repeated.
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13
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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
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OFFSET
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0,3
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COMMENTS
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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
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LINKS
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Table of n, a(n) for n=0..33.
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FORMULA
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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
Fom Reinhard Zumkeller, Nov 14 2014: (Start)
a(n) = A000984(floor(n/2)).
a(n) = A249095(n,n) = A249308(n) / 2^n. (End)
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MATHEMATICA
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(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 *)
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PROG
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(Haskell)
a128014 = a000984 . flip div 2
-- Reinhard Zumkeller, Nov 14 2014
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CROSSREFS
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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
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KEYWORD
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easy,nonn
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AUTHOR
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Paul Barry, Feb 11 2007
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STATUS
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approved
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