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a(n) = Sum_{k=0..floor(n/2)} Catalan(k).
3

%I #22 Mar 11 2020 22:58:27

%S 1,1,2,2,4,4,9,9,23,23,65,65,197,197,626,626,2056,2056,6918,6918,

%T 23714,23714,82500,82500,290512,290512,1033412,1033412,3707852,

%U 3707852,13402697,13402697,48760367,48760367,178405157,178405157,656043857

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

%C Hankel transform is A166446(n+2). - _Paul Barry_, Jun 23 2010

%H Andrei Asinowski, Cyril Banderier, Valerie Roitner, <a href="https://lipn.univ-paris13.fr/~banderier/Papers/several_patterns.pdf">Generating functions for lattice paths with several forbidden patterns</a>, (2019).

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

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

%F 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

%F 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

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

%Y Cf. A000108, A014137.

%K easy,nonn

%O 0,3

%A _Paul Barry_, Jul 15 2005