%I A138356
%S A138356 1,1,2,4,10,27,82,268,940,3476,13448,53968,223412,949535,4128594,
%T A138356 18310972,82645012,378851428,1760998280,8288679056,39457907128,
%U A138356 189784872428,921472827272,4512940614960,22279014978544,110797225212112
%N A138356 Moment sequence of t^2 coefficient in det(tI-A) for random matrix A in 
               USp(4).
%C A138356 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 A138356 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 A138356 a(n) is the nth moment of the coefficient of t^2 in L_p(t/sqrt(p)), as 
               p varies.
%C A138356 See A095922 for central moments.
%D A138356 Kiran S. Kedlaya and Andrew V. Sutherland, "Hyperelliptic curves, L-polynomials 
               and random matrices", preprint, 2008.
%D A138356 Kiran S. Kedlaya and Andrew V. Sutherland "Computing L-series of hyperelliptic 
               curves", Algorithmic Number Theory Symposium--ANTS VIII, 2008.
%D A138356 Nicholas M. Katz and Peter Sarnak, "Random Matrices, Frobenius Eigenvalues 
               and Monodromy", AMS, 1999.
%F A138356 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. a(n)=Sum_{i=0..n}binomial(n,i)2^{n-i}*(A126120(i)A126120(i+2)-A126120(i+\
               1)^2).
%e A138356 a(3) = 4 because E[X^3] = 4 for X the t^2 coeff of det(tI-A) in USp(4).
%e A138356 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)
%e A138356 = 8 - 12 + 12 - 4 = 4.
%Y A138356 Cf. A095922, A138349.
%Y A138356 Sequence in context: A148106 A099950 A121690 this_sequence A148107 A148108 
               A057786
%Y A138356 Adjacent sequences: A138353 A138354 A138355 this_sequence A138357 A138358 
               A138359
%K A138356 nonn
%O A138356 0,3
%A A138356 Andrew V. Sutherland (drew(AT)math.mit.edu), Mar 17 2008