%I #28 Mar 31 2014 07:33:20
%S 1,2,9,57,411,3181,25803,216486,1863139,16356925,145914573,1318844414,
%T 12051758083,111159508991,1033505202643,9675905948106,91140492185703,
%U 863107104436546,8212873185281571,78484928498979435,752928813642151089
%N Number of lattice paths from (0,0) to (n,n) using steps S={(k,0),(0,k),(1,1)|k>0} which never go above the line y=x.
%H Vincenzo Librandi, <a href="/A175912/b175912.txt">Table of n, a(n) for n = 0..200</a>
%H J. P. S. Kung and A. de Mier, <a href="http://dx.doi.org/10.1016/j.jcta.2012.08.010">Catalan lattice paths with rook, bishop and spider steps</a>, Journal of Combinatorial Theory, Series A 120 (2013) 379-389. - From _N. J. A. Sloane_, Dec 27 2012
%H Joseph P. S. Kung, Anna de Mier, <a href="http://arxiv.org/abs/1109.1806">Rook and queen paths with boundaries</a>, arXiv:1109.1806.
%F Asymptotic: a(n) ~ b*c^n/n^(3/2), where c = 10.33185141266662366... is the root of the equation c^3-11*c^2+7*c-1=0 and b = sqrt(13*c-5-c^2)*(2*c^2+9*c-2)/(2*c^3*sqrt(Pi)) = 0.36996178... - _Vaclav Kotesovec_, Dec 25 2013
%F G.f. (from reference): (1+2*x-x^2 - sqrt((x-1)*(x^3-7*x^2+11*x-1)))/(2*x*(x-2)^2). - _Vaclav Kotesovec_, Dec 25 2013
%t Flatten[{1,RecurrenceTable[{(n+1)*a[n]-10*(n+2)*a[n+1]+(34*n+96)*a[n+2]-6*(8*n+29)*a[n+3]+5*(5*n+23)*a[n+4]-2*(n+6)*a[n+5]==0, a[1]==2, a[2]==9, a[3]==57, a[4]==411, a[5]==3181},a,{n,20}]}] (* _Vaclav Kotesovec_, Sep 07 2012 *)
%Y Cf. A006318, A175895, A175896, A175900.
%K nonn
%O 0,2
%A _Eric Werley_, Dec 05 2010
%E Minor edits _Vaclav Kotesovec_, Mar 31 2014