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A138551
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Moment sequence of t^3 coefficient in det(tI-A) for random matrix A in USp(6).
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1
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1, 0, 2, 0, 23, 0, 684, 0, 34760, 0, 2493096, 0, 228253267, 0, 25091028820, 0, 3179942075960, 0, 451649016238160, 0, 70421753109861592, 0, 11869050034269797984, 0, 2136758627313217104448, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Let the random variable X be the coefficient of t^3 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 n-th moment of the coefficient of t^3 in L_p(t/sqrt(p)), as p varies.
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REFERENCES
| Kiran S. Kedlaya and Andrew V. Sutherland, "Hyperelliptic curves, L-polynomials and random matrices", preprint, 2008.
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FORMULA
| See Prop. 12 of Kedlaya-Sutherland reference below.
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EXAMPLE
| a(4) = 23 because E[X^4] = 23 for X the t^3 coeff of det(tI-A) in USp(6).
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CROSSREFS
| Cf. 138540, 138549.
Sequence in context: A156438 A009378 A106708 * A133490 A051728 A201954
Adjacent sequences: A138548 A138549 A138550 * A138552 A138553 A138554
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KEYWORD
| nonn
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AUTHOR
| Andrew V. Sutherland (drew(AT)math.mit.edu), Mar 24 2008
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