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A138350
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Moment sequence of tr(A^2) in USp(4).
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2
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1, -1, 3, -6, 20, -50, 175, -490, 1764, -5292, 19404, -60984, 226512, -736164, 2760615, -9202050, 34763300, -118195220, 449141836, -1551580888, 5924217936, -20734762776, 79483257308, -281248448936, 1081724803600, -3863302870000, 14901311070000, -53644719852000
(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(4) (4 X 4 complex matrices which are unitary and symplectic), then a(n)=E[(tr(A^2)^n] is the n-th moment of the trace of A^2. See A138351 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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LINKS
| Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices.
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FORMULA
| a(n)=(1/2)Integral_{x=0..Pi,y=0..Pi}(2cos(2x)+2cos(2y))^n(2cos(x)-2cos(y))^2(2/Pi*sin^2(x))(2/Pi*sin^2(y))dxdy. a(n)=A126120(n)A138364(n+1)-A138364(n)A126120(n+1)
Conjectured e.g.f. BesselI[1,2x](BesselI[0,2x]-BesselI[1,2x])/x. - Benjamin Phillabaum, Feb 25 2011
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EXAMPLE
| a(5) = -50 because E[(tr(A^2))^5] = -50 for a random matrix A in USp(4).
a(5) = A126120(5)A138364(6)-A138364(5)A126120(6) = 0*0-10*5 = -50
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CROSSREFS
| A signed version of A005558, which is the main entry for this sequence.
Cf. A138349, A138351.
Sequence in context: A148573 A148574 A005558 * A148575 A148576 A148577
Adjacent sequences: A138347 A138348 A138349 * A138351 A138352 A138353
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KEYWORD
| sign
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AUTHOR
| Andrew V. Sutherland (drew(AT)math.mit.edu), Mar 16 2008
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