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A024492
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Catalan numbers with odd index: a(n) = binomial(4*n+2, 2*n+1)/(2*n+2).
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21
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1, 5, 42, 429, 4862, 58786, 742900, 9694845, 129644790, 1767263190, 24466267020, 343059613650, 4861946401452, 69533550916004, 1002242216651368, 14544636039226909, 212336130412243110, 3116285494907301262, 45950804324621742364, 680425371729975800390
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OFFSET
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0,2
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COMMENTS
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a(n) and Catalan(n) have the same 2-adic valuation (equal to 1 less than the sum of the digits in the binary representation of (n + 1)). In particular, a(n) is odd iff n is of the form 2^m - 1. -Peter Bala,Aug 02 2016
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LINKS
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FORMULA
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G.f.: (1/2)*x^(-1)*(1-sqrt((1/2)*(1+sqrt(1-16*x)))).
a(n) = 4^n*binomial(2n+1/2, n)/(n+1). -Paul Barry,May 10 2005
a(n) = binomial(4n+1,2n+1)/(n+1). -Paul Barry,Nov 09 2006
a(n) = (1/(2*Pi))*Integral_{x=-2..2} (2+x)^(2*n)*sqrt((2-x)*(2+x)). -Peter Luschny,Sep 12 2011
D-finite with recurrence (n+1)*(2*n+1)*a(n) -2*(4*n-1)*(4*n+1)*a(n-1)=0. -R. J. Mathar,Nov 26 2012
G.f.: (c(sqrt(x)) - c(-sqrt(x)))/(2*sqrt(x)) = (2-(sqrt(1-4*sqrt(x)) + sqrt(1+4*sqrt(x))))/(4*x), with the g.f. c(x) of the Catalan numbersA000108.-Wolfdieter Lang,Feb 23 2014
a(n) = Sum_{k=0..n} (k+1)^2*binomial(2*(n+1),n-k)^2 /(n+1)^2. -Vladimir Kruchinin,Oct 14 2014
G.f.: A(x) = (1/x)*(inverse series of x - 5*x^2 + 8*x^3 - 4*x^4). -Vladimir Kruchinin,Oct 31 2014
a(n) = (1/Pi)*16^(n+1)*Integral_{x=0..Pi/2} (cos x)^(4n+2)*(sin x)^2. -Greg Dresden,May 30 2021
Sum_{n>=0} a(n)/4^n = 2 - sqrt(2). -Amiram Eldar,Mar 16 2022
a(n) = (1/4^n) * Product_{1 <= i <= j <= 2*n} (i + j + 2)/(i + j - 1).
a(n) = Product_{1 <= i <= j <= 2*n} (3*i + j + 2)/(3*i + j - 1). Cf.A000260.(End)
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EXAMPLE
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sqrt((1/2)*(1+sqrt(1-x))) = 1 - (1/8)*x - (5/128)*x^2 - (42/2048)*x^3 -...
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MAPLE
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with(combstruct):bin:= {B=Union(Z, Prod(B, B))}: seq (count([B, bin, unlabeled], size=2*n), n=1..18); #Zerinvary Lajos,Dec 05 2007
a:= n -> binomial(4*n+1, 2*n+1)/(n+1):
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MATHEMATICA
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CoefficientList[ Series[1 + (HypergeometricPFQ[{3/4, 1, 5/4}, {3/2, 2}, 16 x] - 1), {x, 0, 17}], x]
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PROG
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(MuPAD) combinat::dyckWords::count(2*n+1)$ n = 0..24 //Zerinvary Lajos,Jul 02 2008
(Magma) [Factorial(4*n+2)/(Factorial(2*n+1)*Factorial(2*n+2)): n in [0..20]]; //Vincenzo Librandi,Sep 13 2011
(Maxima) a(n):=sum((k+1)^2*binomial(2*(n+1), n-k)^2, k, 0, n)/(n+1)^2; /*Vladimir Kruchinin,Oct 14 2014 */
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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