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Search: id:A138549
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| A138549 |
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Moment sequence of t^2 coefficient in det(tI-A) for random matrix A in USp(6). |
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+0
2
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| 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
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0,3
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COMMENT
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Let the random variable X be the coefficient of t^2 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].
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 nth moment of the coefficient of t^2 in L_p(t/sqrt(p)), as p varies.
See A138550 for central moments.
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REFERENCES
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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.
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FORMULA
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See Prop. 12 of first Kedlaya-Sutherland reference below.
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EXAMPLE
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a(3) = 5 because E[X^3] = 5 for X the t^2 coeff of det(tI-A) in USp(6).
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CROSSREFS
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Cf. 138540, 138550, 138356.
Sequence in context: A124531 A129578 A005387 this_sequence A144188 A157314 A159603
Adjacent sequences: A138546 A138547 A138548 this_sequence A138550 A138551 A138552
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KEYWORD
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nonn
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AUTHOR
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Andrew V. Sutherland (drew(AT)math.mit.edu), Mar 24 2008
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