A292361 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.

1, 3, 21, 192, 2009, 22818, 273895, 3421318, 44042729, 580473551, 7796745921, 106365396629, 1470068855112, 20543335134692, 289818595800636, 4122517765350669, 59066177091706608

Offset: 0

Author

Timothy Budd, Sep 15 2017

Keywords

• nonn
• walk

Links

• T. Budd, Winding of simple walks on the square lattice, arXiv:1709.04042 [math.CO], 2017.

Crossrefs

Cf. A135404.

Programs

Mathematica

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}]

Formula

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).