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A138540
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Moment sequence of tr(A) in USp(6).
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7
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1, 0, 1, 0, 3, 0, 15, 0, 104, 0, 909, 0, 9449, 0, 112398, 0, 1489410, 0, 21562086, 0, 336086022, 0, 5577242292, 0, 97671172836, 0, 1792348213025, 0, 34268124834495, 0, 679376016769260, 0, 13911118850603610, 0, 293220749128031010, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| If A is a random matrix in the compact group USp(6) (6x6 complex matrices which are unitary and symplectic), then a(n)=E[(tr(A))^n] is the n-th moment of the trace of A.
The multiplicity of the trivial representation in the n-th tensor power of the standard representation of USp(6).
Number of returning walks of length n on a cubic lattice remaining in the chamber x >= y >= z >= 0.
Under a generalized Sato-Tate conjecture, this is the moment sequence of the distribution of unitarized Frobenius traces a_p/sqrt(p) (as p varies), for almost all genus 3 curves.
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REFERENCES
| Kiran S. Kedlaya and Andrew V. Sutherland, "Hyperelliptic curves, L-polynomials and random matrices", preprint, 2008.
David J. Grabiner and Peter Magyar, "Random walks in Weyl chambers and the decomposition of tensor powers", Journal of Algebraic Combinatorics, vol. 2 (1993), no. 3, pp 239-260.
Nicholas M. Katz and Peter Sarnak, "Random Matrices, Frobenius Eigenvalues and Monodromy", AMS, 1999.
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LINKS
| Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices.
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FORMULA
| mgf is A(z) = det[F_{i+j-2}(z)], 1<=i,j<=g, where F_m(z) = Sum_j Binom(m,j)(I_{2j-m}(2z)-I_{2j-m+2}) and I_k(z) is the hyperbolic Bessel function (of the first kind) of order k.
mgf: A(z) = det[F_{i+j-2}(z)], 1<=i,j<=3, where F_m(z) = Sum_j Binom(m,j)(I_{2j-m}(2z)-I_{2j-m+2}(2z)) and I_k(z) is the hyperbolic Bessel function (of the first kind) of order k.
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EXAMPLE
| a(4)=3 because E[(tr(A)^4] = 3 for a random matrix A in USp(6).
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CROSSREFS
| Cf. 138349.
Sequence in context: A135399 A065121 A167339 * A123023 A130637 A054882
Adjacent sequences: A138537 A138538 A138539 * A138541 A138542 A138543
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KEYWORD
| nonn
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AUTHOR
| Andrew V. Sutherland (drew(AT)math.mit.edu), Mar 24 2008, Apr 01 2008
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