%I #26 Mar 31 2014 07:39:59
%S 1,2,10,62,448,3495,28640,242946,2114829,18783658,169546150,
%T 1550728135,14340859992,133867779775,1259689173181,11936488052113,
%U 113799596287017,1090803942244627,10505978544362607,101623141479156708,986801698075230291
%N Number of lattice paths from (0,0) to (n,n) using steps S={(k,0),(0,k),(r,r)|k>0,0<r<=2} which never go above the line y=x.
%D J. P. S. Kung and A. de Mier, Catalan lattice paths with rook, bishop and spider steps, Journal of Combinatorial Theory, Series A 120 (2013) 379-389. - From _N. J. A. Sloane_, Dec 27 2012
%H Vincenzo Librandi, <a href="/A175939/b175939.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) ~ b*c^n/n^(3/2), where c = 10.458904071481665... is the root of the equation x^4-10*x^3-5*x^2+2*x+1=0 and b = sqrt(2*(1-5*c-15*c^2+2*c^3) /c^3)*(-5 - 4*c + 21*c^2 + 27*c^3) / (44*c^3*sqrt(Pi)) = 0.3791408579... - _Vaclav Kotesovec_, Aug 10 2013
%F G.f.: ((x^4+2*x^3-5*x^2-10*x+1)^(1/2)-x^2-3*x-1)/(2*x*(x-1)*(x+2)^2). - _Mark van Hoeij_, Apr 16 2013
%t Flatten[{1,RecurrenceTable[{(n+2)*a[n]+(3*n+7)*a[n+1]-(5*n+24)*a[n+2]-(19*n+90)*a[n+3]+(n+3)*a[n+4]+(21*n+116)*a[n+5]-2*(n+7)*a[n+6]==0, a[1]==2, a[2]==10, a[3]==62, a[4]==448, a[5]==3495, a[6]==28640},a,{n,20}]}] (* _Vaclav Kotesovec_, Sep 07 2012 *)
%Y Cf. A175934, A175935, A175936, A175937.
%K nonn
%O 0,2
%A _Eric Werley_, Dec 06 2010
%E Minor edits _Vaclav Kotesovec_, Mar 31 2014