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%I #29 Jan 16 2024 11:45:36
%S 1,0,1,1,5,16,75,366,2016,11936,75678,507575,3575693,26289408,
%T 200709665,1584482382,12888498820,107698656192,922140333952,
%U 8072379904752,72108967554160,656190909218560,6074106708205200,57118680813847840
%N Central moment sequence of t^2 coefficient in det(tI-A) for random matrix A in USp(6).
%C Let the random variable X be the coefficient of t^2 in the characteristic polynomial det(tI-A) of a random matrix in USp(6) (6x6 complex matrices that are unitary and symplectic). Then a(n) = E[(X-1)^n] is the n-th central moment of X since E[X]=1 (see A138549).
%C Dimension of space of invariant tensors in second fundamental representation of Sp(6). - _Bruce Westbury_, Dec 05 2014
%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 a(n) = Sum_{k=0..n} binomial(n,k)(-1)^{n-k}*A138549(k).
%e a(4) = 5 because E[(X-1)^4] = 5 for X the t^2 coeff of det(tI-A) in USp(6).
%o (LiE) p_tensor(n,[0,1,0],C3)|[0,0,0]
%Y Cf. A138540, A138549, A251591.
%K nonn
%O 0,5
%A _Andrew V. Sutherland_, Mar 24 2008