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A138552
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Returning walks of length 2n on the upper half of the square lattice, distinct under reflections about the y-axis.
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0
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1, 2, 11, 90, 889, 9723, 113322, 1380522, 17382365, 224573349, 2962117366, 39741658047, 540862505806, 7450655906450, 103713126384420, 1456845308244810, 20627719676855685, 294136002612344145
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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FORMULA
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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
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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