|
%I #11 Feb 01 2016 09:36:33
%S 1,2,5,16,51,180,622,2288,8227,30940,113882,434112,1622414,6240360,
%T 23572668,91245600,347398435,1351035180,5174839714,20197293600,
%U 77729205658,304232711640,1175332659332,4610721207456,17868732968846,70228114687640,272886854006852
%N Number of paths from (0,0) to (n,n) that use E(1,0) and N(0,1) as steps and have even number of East steps below the line y=x-1.
%H Ran Pan and Jeffrey B. Remmel, <a href="http://arxiv.org/abs/1601.07988">Paired patterns in lattice paths</a>, arXiv:1601.07988 [math.CO], 2016.
%F G.f.: -(-1+f(x)+2*x)*(4+1/f(x)+2*f(x)+Sqrt(1+4*x))/(16*x^2), where f(x)=Sqrt(1-4*x).
%t CoefficientList[Series[-(-1 + Sqrt[1 - 4 x] + 2 x) (4 + 1/(Sqrt[1 - 4 x]) + 2 Sqrt[1 - 4 x] + Sqrt[1 + 4 x])/(16 x^2), {x, 0, 33}], x] (* _Vincenzo Librandi_ Feb 01 2016 *)
%o (PARI) x='x+O('x^50); f=sqrt(1-4*x); Vec((4+1/f+sqrt(1+4*x)+2*f)*(1-f-2*x)/(16*x^2)) \\ _Charles R Greathouse IV_, Feb 01 2016
%Y Cf. A000108, A268214.
%K nonn
%O 0,2
%A _Ran Pan_, Jan 28 2016
%E Typo in Gf fixed by _Vincenzo Librandi_, Feb 01 2016
|
