OFFSET
0,2
LINKS
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
KEYWORD
nonn,walk
AUTHOR
Timothy Budd, Sep 15 2017
STATUS
approved