OFFSET
0,3
COMMENTS
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
For n >= 1, number of ballot sequences of length n-1 in which the vote is tied or decided by 1 vote. - Nachum Dershowitz, Aug 12 2020 [edited by Peter Munn, Jan 04 2026]
Number of binary strings of length n that are abelian squares. - Michael S. Branicky, Dec 21 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)).
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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Feb 11 2007
EXTENSIONS
Edited by Peter Munn, Jan 04 2026
STATUS
approved