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

0,2

Links

Table of n, a(n) for n=0..16.

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

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).

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}]

Crossrefs

Cf. A135404.

Sequence in context: A212072 A212029 A219535 * A369783 A151388 A210670

Adjacent sequences: A292358 A292359 A292360 * A292362 A292363 A292364

Keyword

nonn,walk

Author

Timothy Budd, Sep 15 2017

Status

approved