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Search: id:A138543
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| A138543 |
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Moment sequence of tr(A^3) in USp(6). |
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+0
2
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| 1, 0, 3, 0, 26, 0, 345, 0, 5754, 0, 110586, 0, 2341548, 0, 53208441, 0, 1276027610, 0, 31930139670, 0, 826963069140, 0, 22035414489270, 0, 601361536493340, 0, 16749316314679500, 0, 474777481850283240, 0, 13665774112508864385, 0
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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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^3))^n] is the nth
moment of the trace of A^3.
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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.
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FORMULA
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mgf is A(z) = det[F_{i+j-2}(z)], 1<=i,j<=3, where F_m(z) = Sum_j Binom(m,j)(B_{(2j-m)/3}(z)-B_{(2j-m+2)/3}(z)) and B_v(z)=0 for non-integer v and otherwise B_v(z)=I_v(2z) with I_v(z) is the hyperbolic Bessel function (of the first kind) of order v.
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EXAMPLE
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a(4) = 26 because E[(tr(A^2))^4] = 26 for a random matrix A in USp(6).
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CROSSREFS
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Cf. 138540.
Sequence in context: A013388 A057379 A009777 this_sequence A007415 A058833 A012775
Adjacent sequences: A138540 A138541 A138542 this_sequence A138544 A138545 A138546
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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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