A268213 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.

1, 2, 5, 16, 51, 180, 622, 2288, 8227, 30940, 113882, 434112, 1622414, 6240360, 23572668, 91245600, 347398435, 1351035180, 5174839714, 20197293600, 77729205658, 304232711640, 1175332659332, 4610721207456, 17868732968846, 70228114687640, 272886854006852

Offset

0,2

Links

Table of n, a(n) for n=0..26.

Ran Pan and Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.

Formula

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).

Mathematica

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 *)

Prog

(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

Crossrefs

Cf. A000108, A268214.

Sequence in context: A148388 A148389 A108529 * A231357 A303477 A234843

Adjacent sequences: A268210 A268211 A268212 * A268214 A268215 A268216

Keyword

nonn

Author

Ran Pan, Jan 28 2016

Extensions

Typo in Gf fixed by Vincenzo Librandi, Feb 01 2016

Status

approved