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Search: id:A138544
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| A138544 |
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Moment sequence of tr(A^4) in USp(6). |
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+0
2
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| 1, -1, 4, -9, 42, -130, 660, -2415, 12810, -51786, 281736, -1216446, 6727644, -30440124, 170316432, -798126615, 4504487130, -21692469370, 123255492360, -606672653730, 3465702008340, -17366224451940, 99645553785960, -506814533253210, 2918768920720380, -15034038412333500
(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^4))^n] is the nth
moment of the trace of A^4. See A138545 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.
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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)/4}(z)-B_{(2j-m+2)/4}(z)) and B_v(z)=0 for non-integer v and otherwise B_v(z)=I_v(2z), with I_v(z) the hyperbolic Bessel function (of the first kind) of order v.
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EXAMPLE
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a(3) = -9 because E[(tr(A^4))^3] = -9 for a random matrix A in USp(6).
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CROSSREFS
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Cf. A138540, A138545.
Sequence in context: A149167 A149168 A149169 this_sequence A093149 A048054 A149170
Adjacent sequences: A138541 A138542 A138543 this_sequence A138545 A138546 A138547
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KEYWORD
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sign
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AUTHOR
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Andrew V. Sutherland (drew(AT)math.mit.edu), Mar 24 2008
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