A292361 - OEIS

%I #9 Oct 09 2017 23:33:46

%S 1,3,21,192,2009,22818,273895,3421318,44042729,580473551,7796745921,

%T 106365396629,1470068855112,20543335134692,289818595800636,

%U 4122517765350669,59066177091706608

%N The number of paths of length 2m in the plane, starting and ending at (0,1), with unit steps in the four directions (north, east, south, west) and staying in the region y > 0 or x > -y.

%H T. Budd, <a href="https://arxiv.org/abs/1709.04042">Winding of simple walks on the square lattice</a>, arXiv:1709.04042 [math.CO], 2017.

%F G.f.: A(x) = 1/(2x) - (Pi / (4 x K(16x))) * (1 + 2 Sum_{n>=1} (q^n + 3q^(2n)+ q^(3n)) / (1 + q^n + q^(2n) + q^(3n) + q^(4n)) ), where q=q(16x) is the Jacobi nome of parameter m=16x and K(16x) is the complete elliptic integral of the first kind of parameter m=16x (proven).

%t a[n_] := SeriesCoefficient[-Pi(1 + 2 Sum[(y+3y^2+y^3)/(1+y+y^2+y^3+y^4) /. y->EllipticNomeQ[m]^l, {l,n+1}])/(4EllipticK[m]) /. m->16x, {x,0,n+1}]

%Y Cf. A135404.

%K nonn,walk

%O 0,2

%A _Timothy Budd_, Sep 15 2017