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A138543 Moment sequence of tr(A^3) in USp(6). 1

%I #16 Jul 25 2019 03:11:27

%S 1,0,3,0,26,0,345,0,5754,0,110586,0,2341548,0,53208441,0,1276027610,0,

%T 31930139670,0,826963069140,0,22035414489270,0,601361536493340,0,

%U 16749316314679500,0,474777481850283240,0,13665774112508864385,0

%N Moment sequence of tr(A^3) in USp(6).

%C If A is a random matrix in the compact group USp(6) (6 X 6 complex matrices which are unitary and symplectic), then a(n) = E[(tr(A^3))^n] is the n-th moment of the trace of A^3.

%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)/3}(z)-B_{(2j-m+2)/3}(z)) and B_v(z)=0 for non-integer v and otherwise B_v(z)=I_v(2z) with I_v(z) is the hyperbolic Bessel function (of the first kind) of order v.

%e a(4) = 26 because E[(tr(A^2))^4] = 26 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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