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 A024492 Catalan numbers with odd index: a(n) = binomial(4*n+2, 2*n+1)/(2*n+2). 16

%I

%S 1,5,42,429,4862,58786,742900,9694845,129644790,1767263190,

%T 24466267020,343059613650,4861946401452,69533550916004,

%U 1002242216651368,14544636039226909,212336130412243110,3116285494907301262

%N Catalan numbers with odd index: a(n) = binomial(4*n+2, 2*n+1)/(2*n+2).

%C 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

%H Vincenzo Librandi, <a href="/A024492/b024492.txt">Table of n, a(n) for n = 0..100</a>

%F G.f.: A(x) = 1/2*x^-1*(1-sqrt(1/2*(1+sqrt(1-16*x)))).

%F G.f.: 3F2([3/4, 1, 5/4], [3/2, 2], 16*x). - _Olivier GĂ©rard_, Feb 16 2011

%F a(n) = 4^n*binomial(2n+1/2, n)/(n+1). - _Paul Barry_, May 10 2005

%F a(n) = binomial(4n+1,2n+1)/(n+1). - _Paul Barry_, Nov 09 2006

%F a(n) = (1/(2*Pi)*integral(x=-2..2, (2+x)^(2*n)*sqrt((2-x)*(2+x))). - _Peter Luschny_, Sep 12 2011

%F (n+1)*(2*n+1)*a(n) -2*(4*n-1)*(4*n+1)*a(n-1)=0. - _R. J. Mathar_, Nov 26 2012

%F 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 numbers A000108. - _Wolfdieter Lang_, Feb 23 2014

%F a(n) = sum(k=0..n, (k+1)^2*binomial(2*(n+1),n-k)^2)/(n+1)^2. - _Vladimir Kruchinin_, Oct 14 2014

%F G.f.: A(x) = (1/x)*(inverse series of x - 5*x^2 + 8*x^3 - 4*x^4). - _Vladimir Kruchinin_, Oct 31 2014

%F a(n) ~ sqrt(2)*16^n/(sqrt(Pi)*n^(3/2)). - _Ilya Gutkovskiy_, Aug 02 2016

%e sqrt(1/2*(1+sqrt(1-x))) = 1 - 1/8*x - 5/128*x^2 - 42/2048*x^3 - ...

%p 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

%t CoefficientList[ Series[1 + (HypergeometricPFQ[{3/4, 1, 5/4}, {3/2, 2}, 16 x] - 1), {x, 0, 17}], x]

%t CatalanNumber[Range[1,41,2]] (* _Harvey P. Dale_, Jul 25 2011 *)

%o (MuPAD) combinat::catalan(2*n+1)\$ n = 0..24 // _Zerinvary Lajos_, Jul 02 2008

%o (MuPAD) combinat::dyckWords::count(2*n+1)\$ n = 0..24 // _Zerinvary Lajos_, Jul 02 2008

%o (MAGMA) [Factorial(4*n+2)/(Factorial(2*n+1)*Factorial(2*n+2)): n in [0..20]]; // _Vincenzo Librandi_, Sep 13 2011

%o (PARI) a(n)=binomial(4*n+2,2*n+1)/(2*n+2) \\ _Charles R Greathouse IV_, Sep 13 2011

%o (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 */

%Y Cf. A048990 (Catalan numbers with even index), A024491, A000108, A000894, A228329.

%K nonn,easy,nice

%O 0,2

%A _Clark Kimberling_

%E More terms from _Wolfdieter Lang_

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Last modified January 23 17:24 EST 2019. Contains 319399 sequences. (Running on oeis4.)