The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A048990 Catalan numbers with even index (A000108(2*n), n >= 0): a(n) = binomial(4*n, 2*n)/(2*n+1). 18
 1, 2, 14, 132, 1430, 16796, 208012, 2674440, 35357670, 477638700, 6564120420, 91482563640, 1289904147324, 18367353072152, 263747951750360, 3814986502092304, 55534064877048198, 812944042149730764, 11959798385860453492, 176733862787006701400 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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 Self-convolution of a(n)/4^n gives Catalan numbers (A000108). - Vladimir Reshetnikov, Oct 10 2016 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 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 Paul Barry, Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials, arXiv:1910.00875 [math.CO], 2019. Gi-Sang Cheon, S.-T. Jin, and L. W. Shapiro, A combinatorial equivalence relation for formal power series, Linear Algebra and its Applications, Vol. 491 (2016), pp. 123-137. Greg Markowsky, A method for deriving hypergeometric and related identities from the H 2 Hardy norm of conformal maps, Integral Transforms and Special Functions, Vol. 24, No. 4 (2013), pp. 302-313; arXiv preprint, arXiv:1205.2458 [math.CV], 2012. Khodabakhsh Hessami Pilehrood and Tatiana Hessami Pilehrood, Jacobi Polynomials and Congruences Involving Some Higher-Order Catalan Numbers and Binomial Coefficients, J. Int. Seq., Vol. 18 (2015), Article 15.11.7. FORMULA a(n) = 2 * A065097(n) - A000007(n). G.f.: A(x) = sqrt((1/8)*x^(-1)*(1-sqrt(1-16*x))). G.f.: 2F1( (1/4, 3/4); (3/2))(16*x). - Olivier Gérard Feb 17 2011 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 E.g.f: 2F2(1/4, 3/4; 1, 3/2; 16*x). - Vladimir Reshetnikov, Apr 24 2013 G.f. A(x) satisfies: A(x) = exp( x*A(x)^4 + Integral(A(x)^4 dx) ). - Paul D. Hanna, Nov 09 2013 G.f. A(x) satisfies: A(x) = sqrt(1 + 4*x*A(x)^4). - Paul D. Hanna, Nov 09 2013 a(n) = hypergeom([1-2*n,-2*n],[2],1). - Peter Luschny, Sep 22 2014 a(n) ~ 2^(4*n-3/2)/(sqrt(Pi)*n^(3/2)). - Ilya Gutkovskiy, Oct 10 2016 From Peter Bala, Feb 27 2020: (Start) a(n) = (4^n)*binomial(2*n + 1/2, n)/(4*n + 1). 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) Sum_{n>=0} 1/a(n) = A276483. - Amiram Eldar, Nov 18 2020 Sum_{n>=0} a(n)/4^n = sqrt(2). - Amiram Eldar, Mar 16 2022 EXAMPLE sqrt(2*x^-1*(1-sqrt(1-x))) = 1 + (1/8)*x + (7/128)*x^2 + (33/1024)*x^3 + ... MATHEMATICA f[n_] := CatalanNumber[ 2n]; Array[f, 18, 0] (* Or *) CoefficientList[ Series[ Sqrt[2]/Sqrt[1 + Sqrt[1 - 16 x]], {x, 0, 17}], x] (* Robert G. Wilson v *) CatalanNumber[Range[0, 40, 2]] (* Harvey P. Dale, Mar 19 2015 *) PROG (MuPAD) combinat::dyckWords::count(2*n) \$ n = 0..28 // Zerinvary Lajos, Apr 14 2007 (PARI) /* G.f.: A(x) = exp( x*A(x)^4 + Integral(A(x)^4 dx) ): */ {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 for(n=0, 30, print1(a(n), ", ")) (Sage) A048990 = lambda n: hypergeometric([1-2*n, -2*n], [2], 1) [Integer(A048990(n).n()) for n in range(20)] # Peter Luschny, Sep 22 2014 CROSSREFS Cf. A000007, A000108, A024492, A065097, A099250, A228411, A276483. Sequence in context: A235352 A146971 A246481 * A089602 A336960 A052641 Adjacent sequences:  A048987 A048988 A048989 * A048991 A048992 A048993 KEYWORD easy,nonn AUTHOR STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified May 23 23:07 EDT 2022. Contains 353993 sequences. (Running on oeis4.)