A110199 a(n) = Sum_{k=0..floor(n/2)} Catalan(k).

1, 1, 2, 2, 4, 4, 9, 9, 23, 23, 65, 65, 197, 197, 626, 626, 2056, 2056, 6918, 6918, 23714, 23714, 82500, 82500, 290512, 290512, 1033412, 1033412, 3707852, 3707852, 13402697, 13402697, 48760367, 48760367, 178405157, 178405157, 656043857

Offset: 0

Author

Paul Barry, Jul 15 2005

Keywords

• easy
• nonn

Comments

Hankel transform is A166446(n+2). - Paul Barry, Jun 23 2010

Links

• Andrei Asinowski, Cyril Banderier, Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).

Crossrefs

Cf. A000108, A014137.

Programs

Maple

a:= n-> add(binomial(2*j, j)/(j+1), j=0..n/2): seq(a(n), n=0..36); # Zerinvary Lajos, Apr 30 2007

Formula

G.f.: (1-sqrt(1-4x^2))/((1-x)2x^2);

a(n) = Sum_{k=0..floor(n/2)} binomial(2k, k)/(k+1).

Conjecture: -(n+2)*a(n) + (n+2)*a(n-1) + 4*(n-1)*a(n-2) + 4*(1-n)*a(n-3) = 0. - R. J. Mathar, Nov 09 2012

G.f.: 1/x^2 - G(0)/(1-x)/x^2, where G(k)= 1 - x/(1 - x/(1 + x/(1 + x/G(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Jul 17 2013