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%I #43 Apr 24 2023 13:39:04
%S 1,1,2,2,6,6,20,20,70,70,252,252,924,924,3432,3432,12870,12870,48620,
%T 48620,184756,184756,705432,705432,2704156,2704156,10400600,10400600,
%U 40116600,40116600,155117520,155117520,601080390,601080390
%N Central binomial coefficients C(2n,n) repeated.
%C Binomial transform is A097893. Hankel transform is A128017.
%C Hankel transform of a(n+1) is A128018. - _Paul Barry_, Nov 23 2009
%C Number of 2n-bead balanced binary necklaces that are equivalent to their reverse. - _Andrew Howroyd_, Sep 29 2017
%C Number of ballot sequences of length n in which the vote is tied or decided by 1 vote. - _Nachum Dershowitz_, Aug 12 2020
%C Number of binary strings of length n that are abelian squares. - _Michael S. Branicky_, Dec 21 2020
%F G.f.: (1+x)/sqrt(1-4*x^2).
%F a(n) = C(n,n/2)*(1+(-1)^n)/2 + C(n-1,(n-1)/2)*(1-(-1)^n)/2.
%F a(n) = (1/Pi)*Integral_{x=-2..2} x^n*(1+x)/(x*sqrt(4-x^2)), as moment sequence.
%F E.g.f. of a(n+1): Bessel_I(0,2*x)+2*Bessel_I(1,2*x). - _Paul Barry_, Mar 26 2010
%F 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
%F a(n) = 2^n*Product_{k=0..n-1} ((k/n+1/n)/2)^((-1)^k). - _Peter Luschny_, Dec 03 2013
%F From _Reinhard Zumkeller_, Nov 14 2014: (Start)
%F a(n) = A000984(floor(n/2)).
%F a(n) = A249095(n,n) = A249308(n) / 2^n. (End)
%t (1+x)/Sqrt[1-4x^2] + O[x]^34 // CoefficientList[#, x]& (* _Jean-François Alcover_, Oct 07 2017 *)
%t With[{cb=Table[Binomial[2n,n],{n,0,20}]},Riffle[cb,cb]] (* _Harvey P. Dale_, Feb 17 2020 *)
%o (Haskell)
%o a128014 = a000984 . flip div 2
%o -- _Reinhard Zumkeller_, Nov 14 2014
%Y Cf. A097893, A128017, A128018.
%Y Cf. A000984, A249095, A249308.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Feb 11 2007
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