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Moment sequence of t^2 coefficient in det(tI-A) for random matrix A in USp(4).
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%I #7 May 10 2019 04:33:13

%S 1,1,2,4,10,27,82,268,940,3476,13448,53968,223412,949535,4128594,

%T 18310972,82645012,378851428,1760998280,8288679056,39457907128,

%U 189784872428,921472827272,4512940614960,22279014978544,110797225212112

%N Moment sequence of t^2 coefficient in det(tI-A) for random matrix A in USp(4).

%C Let the random variable X be the coefficient of t^2 in the characteristic polynomial det(tI-A) of a random matrix in USp(4) (4 X 4 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 2 curve C. Under a generalized Sato-Tate conjecture, for almost all C,

%C a(n) is the n-th moment of the coefficient of t^2 in L_p(t/sqrt(p)), as p varies.

%C See A095922 for central moments.

%H Kiran S. Kedlaya, Andrew V. Sutherland, <a href="https://arxiv.org/abs/0801.2778">Computing L-series of hyperelliptic curves</a>, arXiv:0801.2778 [math.NT], 2008-2012; Algorithmic Number Theory Symposium--ANTS VIII, 2008.

%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.

%H Nicholas M. Katz and Peter Sarnak, <a href="http://bookstore.ams.org/coll-45/">Random Matrices, Frobenius Eigenvalues and Monodromy</a>, AMS, 1999.

%F a(n) = (1/2)Integral_{x=0..Pi,y=0..Pi}(4cos(x)cos(y)+2)^n(2cos(x)-2cos(y))^2(2/Pi*sin^2(x))(2/Pi*sin^2(y))dxdy.

%F a(n) = Sum_{i=0..n}binomial(n,i)2^{n-i}*(A126120(i)*A126120(i+2)-A126120(i+1)^2).

%e a(3) = 4 because E[X^3] = 4 for X the t^2 coeff of det(tI-A) in USp(4).

%e a(3) = 1*2^3*(1*1-0^2) + 3*2^2*(0*0-1^2) + 3*2^1*(1*2-0^2) + 1*2^0*(0*0-2^2) = 8 - 12 + 12 - 4 = 4.

%Y Cf. A095922, A138349.

%K nonn

%O 0,3

%A _Andrew V. Sutherland_, Mar 17 2008