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 A048990 Catalan numbers with even index (A000108(2*n), n >= 0): a(n) = binomial(4*n, 2*n)/(2*n+1). 14

%I

%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

%H Vincenzo Librandi, <a href="/A048990/b048990.txt">Table of n, a(n) for n = 0..200</a>

%H G. Markowsky, <a href="http://arxiv.org/abs/1205.2458">A method for deriving hypergeometric and related identities from the H^2 Hardy norm of conformal maps</a>, arXiv preprint arXiv:1205.2458 [math.CV], 2012.

%H Gi-Sang Cheon, S.-T. Jin, 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, Available online 30 March 2015.

%H K. H. Pilehrood, T. H. 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. 18 (2015) 15.11.7

%F a(n) = 2 * A065097(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 GĂ©rard_ Feb 17 2011

%F 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

%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. A000108, A024492, A065097.

%K easy,nonn

%O 0,2

%A _Wolfdieter Lang_

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Last modified October 15 11:03 EDT 2018. Contains 316224 sequences. (Running on oeis4.)