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Search: id:A138354
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| A138354 |
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Central moment sequence of tr(A^4) in USp(4). |
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+0
1
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| 1, 0, 3, 1, 21, 26, 215, 498, 2821, 9040, 43695, 165375, 752785, 3101970, 13881803, 59837183, 267860685, 1184749704, 5337504263, 23996776941, 108964583121, 495544446410, 2267450194443, 10402298479276, 47926692348121
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Binomial transform of A018224.
If A is a random matrix in the compact group USp(4) (4 X 4 complex are unitary and symplectic), then a(n)=E[(tr(A^4)+1)^n] is the nth central moment of the trace of A^4, since E[tr(A^4)] = -1 (see A018224).
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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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LINKS
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Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices.
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FORMULA
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a(n)=(1/2)Integral_{x=0..Pi,y=0..Pi}(2cos(4x)+2cos(4y)+1)^n(2cos(x)-2cos(y))^2(2/Pi*sin^2(x))(2/Pi*sin^2(y))dxdy. a(n)=Sum_{i=0..n}binomial(n,i)A018224(i).
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EXAMPLE
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a(3) = 1 because E[(tr(A^4)+1)^3] = 1.
a(3) = 1*A018224(0) + 3*A018224(1) + 3*A018224(2) + 1*A018224(1)
= 1*1 + 3*(-1) + 3*4 + 1*(-9) = 1.
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CROSSREFS
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Cf. A018224.
Sequence in context: A000369 A136236 A113090 this_sequence A010291 A027477 A137330
Adjacent sequences: A138351 A138352 A138353 this_sequence A138355 A138356 A138357
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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 16 2008, Mar 31 2008
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