

A138549


Moment sequence of t^2 coefficient in det(tIA) for random matrix A in USp(6).


2



1, 1, 2, 5, 16, 62, 282, 1459, 8375, 52323, 350676, 2493846, 18659787, 145918295, 1186129168, 9978055080, 86545684565, 771571356565, 7051538798490, 65913863945775, 628919704903746, 6114899366942556, 60492393411513722
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OFFSET

0,3


COMMENTS

Let the random variable X be the coefficient of t^2 in the characteristic polynomial det(tIA) of a random matrix in USp(6) (6x6 complex matrices that are unitary and symplectic). Then a(n) = E[X^n].
Let L_p(T) be the Lpolynomial (numerator of the zeta function) of a genus 3 curve C. Under a generalized SatoTate 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 A138550 for central moments.


LINKS

Table of n, a(n) for n=0..22.
Kiran S. Kedlaya, Andrew V. Sutherland, Computing Lseries of hyperelliptic curves, arXiv:0801.2778 [math.NT], 20082012; Algorithmic Number Theory SymposiumANTS VIII, 2008.
Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, Lpolynomials and random matrices, arXiv:0803.4462 [math.NT], 20082010.
Nicholas M. Katz and Peter Sarnak, Random Matrices, Frobenius Eigenvalues and Monodromy, AMS, 1999.


FORMULA

See Prop. 12 of first KedlayaSutherland reference.


EXAMPLE

a(3) = 5 because E[X^3] = 5 for X the t^2 coeff of det(tIA) in USp(6).


CROSSREFS

Cf. A138540, A138550, A138356.
Sequence in context: A129578 A005387 A173469 * A210667 A144188 A157314
Adjacent sequences: A138546 A138547 A138548 * A138550 A138551 A138552


KEYWORD

nonn


AUTHOR

Andrew V. Sutherland, Mar 24 2008


STATUS

approved



