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%I #119 Feb 24 2023 02:30:26
%S 1,2,14,132,1430,16796,208012,2674440,35357670,477638700,6564120420,
%T 91482563640,1289904147324,18367353072152,263747951750360,
%U 3814986502092304,55534064877048198,812944042149730764,11959798385860453492,176733862787006701400
%N Catalan numbers with even index (A000108(2*n), n >= 0): a(n) = binomial(4*n, 2*n)/(2*n+1).
%C With interpolated zeros, this is C(n)*(1+(-1)^n)/2 with g.f. given by 2/(sqrt(1+4x) + sqrt(1-4x)). - _Paul Barry_, Sep 09 2004
%C Self-convolution of a(n)/4^n gives Catalan numbers (A000108). - _Vladimir Reshetnikov_, Oct 10 2016
%C a(n) is the number of grand Dyck paths from (0,0) to (4n,0) that avoid vertices (2k,0) for all odd k > 0. - _Alexander Burstein_, May 11 2021
%H Vincenzo Librandi, <a href="/A048990/b048990.txt">Table of n, a(n) for n = 0..200</a>
%H Paul Barry, <a href="https://arxiv.org/abs/1910.00875">Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials</a>, arXiv:1910.00875 [math.CO], 2019.
%H Gi-Sang Cheon, S.-T. Jin, and L. W. Shapiro, <a href="http://dx.doi.org/10.1016/j.laa.2015.03.015">A combinatorial equivalence relation for formal power series</a>, Linear Algebra and its Applications, Vol. 491 (2016), pp. 123-137.
%H Greg Markowsky, <a href="https://doi.org/10.1080/10652469.2012.689301">A method for deriving hypergeometric and related identities from the H 2 Hardy norm of conformal maps</a>, Integral Transforms and Special Functions, Vol. 24, No. 4 (2013), pp. 302-313; <a href="http://arxiv.org/abs/1205.2458">arXiv preprint</a>, arXiv:1205.2458 [math.CV], 2012.
%H Khodabakhsh Hessami Pilehrood and Tatiana Hessami Pilehrood, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Pilehrood/pile5.html">Jacobi Polynomials and Congruences Involving Some Higher-Order Catalan Numbers and Binomial Coefficients</a>, J. Int. Seq., Vol. 18 (2015), Article 15.11.7.
%F a(n) = 2 * A065097(n) - A000007(n).
%F G.f.: A(x) = sqrt((1/8)*x^(-1)*(1-sqrt(1-16*x))).
%F G.f.: 2F1( (1/4, 3/4); (3/2))(16*x). - _Olivier Gérard_ Feb 17 2011
%F D-finite with recurrence n*(2*n+1)*a(n) - 2*(4*n-1)*(4*n-3)*a(n-1) = 0. - _R. J. Mathar_, Nov 30 2012
%F E.g.f: 2F2(1/4, 3/4; 1, 3/2; 16*x). - _Vladimir Reshetnikov_, Apr 24 2013
%F G.f. A(x) satisfies: A(x) = exp( x*A(x)^4 + Integral(A(x)^4 dx) ). - _Paul D. Hanna_, Nov 09 2013
%F G.f. A(x) satisfies: A(x) = sqrt(1 + 4*x*A(x)^4). - _Paul D. Hanna_, Nov 09 2013
%F a(n) = hypergeom([1-2*n,-2*n],[2],1). - _Peter Luschny_, Sep 22 2014
%F a(n) ~ 2^(4*n-3/2)/(sqrt(Pi)*n^(3/2)). - _Ilya Gutkovskiy_, Oct 10 2016
%F From _Peter Bala_, Feb 27 2020: (Start)
%F a(n) = (4^n)*binomial(2*n + 1/2, n)/(4*n + 1).
%F O.g.f.: A(x) = sqrt(c(4*x)), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. of the Catalan numbers. Cf. A228411. (End)
%F Sum_{n>=0} 1/a(n) = A276483. - _Amiram Eldar_, Nov 18 2020
%F Sum_{n>=0} a(n)/4^n = sqrt(2). - _Amiram Eldar_, Mar 16 2022
%F From _Peter Bala_, Feb 22 2023: (Start)
%F a(n) = (1/2^(2*n-1)) * Product_{1 <= i <= j <= 2*n-1} (i + j + 2)/(i + j - 1) for n >= 1.
%F a(n) = Product_{1 <= i <= j <= 2*n-1} (3*i + j + 2)/(3*i + j - 1). Cf. A024492. (End)
%e sqrt(2*x^-1*(1-sqrt(1-x))) = 1 + (1/8)*x + (7/128)*x^2 + (33/1024)*x^3 + ...
%t f[n_] := CatalanNumber[ 2n]; Array[f, 18, 0] (* Or *)
%t CoefficientList[ Series[ Sqrt[2]/Sqrt[1 + Sqrt[1 - 16 x]], {x, 0, 17}], x] (* _Robert G. Wilson v_ *)
%t CatalanNumber[Range[0,40,2]] (* _Harvey P. Dale_, Mar 19 2015 *)
%o (MuPAD) combinat::dyckWords::count(2*n) $ n = 0..28 // _Zerinvary Lajos_, Apr 14 2007
%o (PARI) /* G.f.: A(x) = exp( x*A(x)^4 + Integral(A(x)^4 dx) ): */
%o {a(n)=local(A=1+x); for(i=1, n, A=exp(x*A^4 + intformal(A^4 +x*O(x^n)))); polcoeff(A, n)} \\ _Paul D. Hanna_, Nov 09 2013
%o for(n=0, 30, print1(a(n), ", "))
%o (Sage)
%o A048990 = lambda n: hypergeometric([1-2*n,-2*n],[2],1)
%o [Integer(A048990(n).n()) for n in range(20)] # _Peter Luschny_, Sep 22 2014
%Y Cf. A000007, A000108, A024492, A065097, A099250, A187357, A187358, A187359, A228411, A276483.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_
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