Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%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