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%I #10 May 10 2019 04:33:27
%S 1,0,2,0,23,0,684,0,34760,0,2493096,0,228253267,0,25091028820,0,
%T 3179942075960,0,451649016238160,0,70421753109861592,0,
%U 11869050034269797984,0,2136758627313217104448,0
%N Moment sequence of t^3 coefficient in det(tI-A) for random matrix A in USp(6).
%C Let the random variable X be the coefficient of t^3 in the characteristic polynomial det(tI-A) of a random matrix in USp(6) (6x6 complex matrices that are unitary and symplectic). Then a(n) = E[X^n].
%C Let L_p(T) be the L-polynomial (numerator of the zeta function) of a genus 3 curve C. Under a generalized Sato-Tate conjecture, for almost all C, a(n) is the n-th moment of the coefficient of t^3 in L_p(t/sqrt(p)), as p varies.
%H Kiran S. Kedlaya and Andrew V. Sutherland, <a href="http://arXiv.org/abs/0803.4462">Hyperelliptic curves, L-polynomials and random matrices</a>, arXiv:0803.4462 [math.NT], 2008-2010.
%F See Prop. 12 of Kedlaya-Sutherland.
%e a(4) = 23 because E[X^4] = 23 for X the t^3 coeff of det(tI-A) in USp(6).
%Y Cf. A138540, A138549.
%K nonn
%O 0,3
%A _Andrew V. Sutherland_, Mar 24 2008
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