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%I #119 Sep 02 2023 15:44:17
%S 1,5,42,429,4862,58786,742900,9694845,129644790,1767263190,
%T 24466267020,343059613650,4861946401452,69533550916004,
%U 1002242216651368,14544636039226909,212336130412243110,3116285494907301262,45950804324621742364,680425371729975800390
%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>
%H Elżbieta Liszewska and Wojciech Młotkowski, <a href="https://doi.org/10.1016/j.aam.2020.102105">Some relatives of the Catalan sequence</a>, Advances in Applied Mathematics, Vol. 121 (2020), 102105; <a href="https://arxiv.org/abs/1907.10725">arXiv preprint</a>, arXiv:1907.10725 [math.CO], 2019.
%H Pedro J. Miana and Natalia Romero, <a href="https://doi.org/10.1016/j.jnt.2010.01.018">Moments of combinatorial and Catalan numbers</a>, Journal of Number Theory, Vol. 130, No. 8 (August 2010), pp. 1876-1887. See Remark 3, p. 1882.
%F G.f.: (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 Gérard_, 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 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
%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
%F Sum_{n>=0} 1/a(n) = A276484. - _Amiram Eldar_, Nov 18 2020
%F G.f.: A(x) = C(4*x)*C(x*C(4*x)), where C(x) is the g.f. of A000108. - _Alexander Burstein_, May 01 2021
%F a(n) = (1/Pi)*16^(n+1)*Integral_{x=0..Pi/2} (cos x)^(4n+2)*(sin x)^2. - _Greg Dresden_, May 30 2021
%F Sum_{n>=0} a(n)/4^n = 2 - sqrt(2). - _Amiram Eldar_, Mar 16 2022
%F From _Peter Bala_, Feb 22 2023: (Start)
%F a(n) = (1/4^n) * Product_{1 <= i <= j <= 2*n} (i + j + 2)/(i + j - 1).
%F a(n) = Product_{1 <= i <= j <= 2*n} (3*i + j + 2)/(3*i + j - 1). Cf. A000260. (End)
%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
%p a := n -> binomial(4*n+1, 2*n+1)/(n+1):
%p seq(a(n), n=0..17); # _Peter Luschny_, May 30 2021
%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.
%Y Cf. A000260, A006013, A187357, A187358, A187359, A228329, A276484.
%K nonn,easy,nice
%O 0,2
%A _Clark Kimberling_
%E More terms from _Wolfdieter Lang_
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