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A138546 Moment sequence of tr(A^5) in USp(6). 2

%I #17 Jul 25 2019 03:09:26

%S 1,0,4,0,42,0,660,0,12810,0,281736,0,6727644,0,170316432,0,4504487130,

%T 0,123255492360,0,3465702008340,0,99645553785960,0,2918768920720380,0,

%U 86852063374902000,0,2619552500788984200,0,79939673971478231760,0

%N Moment sequence of tr(A^5) in USp(6).

%C If A is a random matrix in the compact group USp(6) (6 X 6 complex matrices that are unitary and symplectic), then a(n) = E[(tr(A^5))^n] is the n-th moment of the trace of A^5.

%H Kiran S. Kedlaya and Andrew V. Sutherland, <a href="http://arXiv.org/abs/0803.4462">Hyperelliptic curves, L-polynomials and random matrices</a>, arXiv:0803.4462 [math.NT], 2008-2010.

%F mgf is A(z) = det[F_{i+j-2}(z)], 1<=i,j<=3, where F_m(z) = Sum_j binomial(m,j)(B_{(2j-m)/5}(z)-B_{(2j-m+2)/5}(z)) and B_v(z)=0 for non-integer v and otherwise B_v(z)=I_v(2z) with I_v(z) the hyperbolic Bessel function (of the first kind) of order v.

%e a(4) = 42 because E[(tr(A^5))^4] = 42 for a random matrix A in USp(6).

%Y Cf. A138540.

%K nonn

%O 0,3

%A _Andrew V. Sutherland_, Mar 24 2008

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