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A138354
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Central moment sequence of tr(A^4) in USp(4).
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0
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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
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OFFSET
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0,3
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COMMENTS
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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 n-th central moment of the trace of A^4, since E[tr(A^4)] = -1 (see A018224).
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LINKS
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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.
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EXAMPLE
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a(3) = 1 because E[(tr(A^4)+1)^3] = 1.
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MATHEMATICA
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a18224[n_] := Binomial[n, Floor[n/2]]^2;
a[n_] := Sum[(-1)^i Binomial[n, i] a18224[i], {i, 0, n}];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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