A024492 Catalan numbers with odd index: a(n) = binomial(4*n+2, 2*n+1)/(2*n+2).

1, 5, 42, 429, 4862, 58786, 742900, 9694845, 129644790, 1767263190, 24466267020, 343059613650, 4861946401452, 69533550916004, 1002242216651368, 14544636039226909, 212336130412243110, 3116285494907301262, 45950804324621742364, 680425371729975800390

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• easy
• nice

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

Examples

			sqrt((1/2)*(1+sqrt(1-x))) = 1 - (1/8)*x - (5/128)*x^2 - (42/2048)*x^3 - ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..100
• Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, Advances in Applied Mathematics, Vol. 121 (2020), 102105; arXiv preprint, arXiv:1907.10725 [math.CO], 2019.
• Pedro J. Miana and Natalia Romero, Moments of combinatorial and Catalan numbers, Journal of Number Theory, Vol. 130, No. 8 (August 2010), pp. 1876-1887. See Remark 3, p. 1882.

Crossrefs

Cf. A048990 (Catalan numbers with even index), A024491, A000108, A000894.

Cf. A000260, A006013, A187357, A187358, A187359, A228329, A276484.

Programs

Magma

[Factorial(4*n+2)/(Factorial(2*n+1)*Factorial(2*n+2)): n in [0..20]]; // Vincenzo Librandi, Sep 13 2011

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
a := n -> binomial(4*n+1, 2*n+1)/(n+1):
seq(a(n), n=0..17); # Peter Luschny, May 30 2021

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 *)

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 */

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

PARI

a(n)=binomial(4*n+2,2*n+1)/(2*n+2) \\ Charles R Greathouse IV, Sep 13 2011

Formula

G.f.: (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) = binomial(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

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

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

Sum_{n>=0} 1/a(n) = A276484. - Amiram Eldar, Nov 18 2020

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

a(n) = (1/Pi)*16^(n+1)*Integral_{x=0..Pi/2} (cos x)^(4n+2)*(sin x)^2. - Greg Dresden, May 30 2021

Sum_{n>=0} a(n)/4^n = 2 - sqrt(2). - Amiram Eldar, Mar 16 2022

From Peter Bala, Feb 22 2023: (Start)

a(n) = (1/4^n) * Product_{1 <= i <= j <= 2*n} (i + j + 2)/(i + j - 1).

a(n) = Product_{1 <= i <= j <= 2*n} (3*i + j + 2)/(3*i + j - 1). Cf. A000260. (End)

Extensions

More terms from Wolfdieter Lang