0,3

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].

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,

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

See A095922 for central moments.

Kiran S. Kedlaya and Andrew V. Sutherland, "Hyperelliptic curves, L-polynomials and random matrices", preprint, 2008.

Kiran S. Kedlaya and Andrew V. Sutherland "Computing L-series of hyperelliptic curves", Algorithmic Number Theory Symposium--ANTS VIII, 2008.

Nicholas M. Katz and Peter Sarnak, "Random Matrices, Frobenius Eigenvalues and Monodromy", AMS, 1999.

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).

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

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.

Cf. A095922, A138349.

Sequence in context: A148106 A099950 A121690 this_sequence A148107 A148108 A057786

Adjacent sequences: A138353 A138354 A138355 this_sequence A138357 A138358 A138359

nonn

Andrew V. Sutherland (drew(AT)math.mit.edu), Mar 17 2008