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%I #14 Jul 25 2019 03:11:40
%S 1,-1,4,-9,42,-130,660,-2415,12810,-51786,281736,-1216446,6727644,
%T -30440124,170316432,-798126615,4504487130,-21692469370,123255492360,
%U -606672653730,3465702008340,-17366224451940,99645553785960,-506814533253210,2918768920720380,-15034038412333500
%N Moment sequence of tr(A^4) 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^4))^n] is the n-th moment of the trace of A^4. See A138545 for central moments.
%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)/4}(z)-B_{(2j-m+2)/4}(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(3) = -9 because E[(tr(A^4))^3] = -9 for a random matrix A in USp(6).
%Y Cf. A138540, A138545.
%K sign
%O 0,3
%A _Andrew V. Sutherland_, Mar 24 2008
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