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A138541
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Moment sequence of tr(A^2) in USp(6).
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1
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1, -1, 3, -7, 24, -75, 285, -1036, 4242, -16926, 73206, -311256, 1403028, -6247527, 29082339, -134138290, 640672890, -3038045010, 14818136190, -71858704710, 356665411440, -1761879027090, 8874875097270, -44526516209280, 227135946200940, -1154738374364100, 5955171596514900
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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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^2))^n] is the nth
moment of the trace of A^2. See A138542 for central moments.
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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
| mgf is A(z) = det[F_{i+j-2}(z)], 1<=i,j<=g, where F_m(z) = Sum_j Binom(m,j)(B_{(2j-m)/2}(z)-B_{(2j-m+2)/2}(z)) and B_v(z)=0 for non-integer k 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
| a(4) = 24 because E[(tr(A^2))^4] = 24 for a random matrix A in USp(6).
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CROSSREFS
| Cf. A138540, A138542.
Sequence in context: A148718 A003449 A148719 * A148720 A038169 A176606
Adjacent sequences: A138538 A138539 A138540 * A138542 A138543 A138544
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KEYWORD
| sign
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AUTHOR
| Andrew V. Sutherland (drew(AT)math.mit.edu), Mar 24 2008
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