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Search: id:A138351
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| A138351 |
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Central moment sequence of tr(A^2) in USp(4). |
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+0
2
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| 1, 0, 2, 1, 11, 16, 95, 232, 1085, 3460, 14820, 54275, 227095, 895688, 3756688, 15462293, 65586405, 277342336, 1192038266, 5136760581, 22357937431, 97730561480, 430177280197, 1901975209706, 8454151507801, 37734802709796
(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(4) (4 X 4 complex matrices which are unitary and symplectic), then a(n)=E[(tr(A^2)+1)^n] is the nth central moment of the trace of A^2, since E[tr(A^2)] = -1 (see A138350).
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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(2x)+2cos(2y)+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)A138350(i)
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EXAMPLE
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a(4) = 11 because E[((tr(A^2)+1)^4] = 11 for a random matrix A in USp(4).
a(4) = 1*A138350(0)+4*A138350(1)+6*A138350(2)+4*A138350(3)+1*A138350(4)
= 1*1 + 4*(-1) + 6*3 + 4*(-6) + 1*20 = 11.
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CROSSREFS
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Cf. A138350.
Sequence in context: A158354 A055459 A080958 this_sequence A120293 A063624 A101851
Adjacent sequences: A138348 A138349 A138350 this_sequence A138352 A138353 A138354
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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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