A138552 Returning walks of length 2n on the upper half of the square lattice, distinct under reflections about the y-axis.

1, 2, 11, 90, 889, 9723, 113322, 1380522, 17382365, 224573349, 2962117366, 39741658047, 540862505806, 7450655906450, 103713126384420, 1456845308244810, 20627719676855685, 294136002612344145

Offset: 0

Author

Andrew V. Sutherland, Mar 24 2008

Keywords

• nonn

Comments

Under reasonable assumptions, a(n)=E[X^{2n}] where the random variable X is the unitarized Frobenius trace X=a_p/sqrt(p) (as p varies) of a genus 2 curve whose Jacobian is isogenous to the product of two elliptic curves, exactly one of which has complex multiplication.

Examples

			a(2) = 11 because EEWW, EWEW, EWWE, EWNS, ENSW, ENWS, NEWS, NESW, NSEW, NSNS, NNSS are all the walks of length 4 on the upper half of the square lattice that are distinct under reflections about the y-axis.
		

Links

• Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2008-2010.

Crossrefs

Cf. A000108, A000891.

Programs

Mathematica

CoefficientList[Series[(3 Pi-2 Pi Sqrt[1-4x]-2EllipticE[16 x])/(8Pi x), {x, 0, 20}], x] (* Benedict W. J. Irwin, Jul 13 2016 *)

Formula

a(n) = (A000891(n) + A000108(n))/2.

G.f.: (3*Pi-2*Pi*sqrt(1-4*x)-2*EllipticE(16*x))/(8*Pi*x). - Benedict W. J. Irwin, Jul 13 2016

a(n) ~ 16^n*n^(-2)/Pi. - Ilya Gutkovskiy, Jul 13 2016

Recurrence: n*(n+1)^2*(3*n - 2)*a(n) = 2*n*(2*n - 1)*(15*n^2 - n - 4)*a(n-1) - 8*(2*n - 3)*(2*n - 1)^2*(3*n + 1)*a(n-2). - Vaclav Kotesovec, Jul 14 2016