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 A024492 Catalan numbers with odd index: a(n) = binomial(4*n+2,2*n+1)/(2*n+2). 16
 1, 5, 42, 429, 4862, 58786, 742900, 9694845, 129644790, 1767263190, 24466267020, 343059613650, 4861946401452, 69533550916004, 1002242216651368, 14544636039226909, 212336130412243110, 3116285494907301262 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..100 FORMULA G.f.: A(x) = 1/2*x^-1*(1-sqrt(1/2*(1+sqrt(1-16*x)))). G.f.: 3F2([3/4, 1, 5/4], [3/2, 2], 16*x). - Olivier Gérard, Feb 16 2011 a(n) = 4^n*binomial(2n+1/2, n)/(n+1). - Paul Barry, May 10 2005 a(n) = C(4n+1,2n+1)/(n+1). - Paul Barry, Nov 09 2006 a(n) = (1/(2*Pi)*integral(x=-2..2, (2+x)^(2*n)*sqrt((2-x)*(2+x))). - Peter Luschny, Sep 12 2011 (n+1)*(2*n+1)*a(n) -2*(4*n-1)*(4*n+1)*a(n-1)=0. - R. J. Mathar, Nov 26 2012 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 a(n) = sum(k=0..n, (k+1)^2*binomial(2*(n+1),n-k)^2)/(n+1)^2. - Vladimir Kruchinin, Oct 14 2014 G.f.: A(x) = (1/x)*(inverse series of x - 5*x^2 + 8*x^3 - 4*x^4). - Vladimir Kruchinin, Oct 31 2014 a(n) ~ sqrt(2)*16^n/(sqrt(Pi)*n^(3/2)). - Ilya Gutkovskiy, Aug 02 2016 EXAMPLE sqrt(1/2*(1+sqrt(1-x))) = 1 - 1/8*x - 5/128*x^2 - 42/2048*x^3 - ... MAPLE 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 MATHEMATICA CoefficientList[ Series[1 + (HypergeometricPFQ[{3/4, 1, 5/4}, {3/2, 2}, 16 x] - 1), {x, 0, 17}], x] CatalanNumber[Range[1, 41, 2]] (* Harvey P. Dale, Jul 25 2011 *) PROG (Mupad) combinat::catalan(2*n+1)\$ n = 0..24 # Zerinvary Lajos, Jul 02 2008 (Mupad) combinat::dyckWords::count(2*n+1)\$ n = 0..24 # Zerinvary Lajos, Jul 02 2008 (MAGMA) [Factorial(4*n+2)/(Factorial(2*n+1)*Factorial(2*n+2)): n in [0..20]]; // Vincenzo Librandi, Sep 13 2011 (PARI) a(n)=binomial(4*n+2, 2*n+1)/(2*n+2) \\ Charles R Greathouse IV, Sep 13 2011 (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 */ CROSSREFS Cf. A048990 (Catalan numbers with even index), A024491, A000108, A000894, A228329. Sequence in context: A082145 A126765 A228793 * A217805 A217808 A151334 Adjacent sequences:  A024489 A024490 A024491 * A024493 A024494 A024495 KEYWORD nonn,easy,nice AUTHOR EXTENSIONS More terms from Wolfdieter Lang STATUS approved

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