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%I #14 Mar 13 2014 07:43:38
%S 1,-1,3,-6,20,-50,175,-490,1764,-5292,19404,-60984,226512,-736164,
%T 2760615,-9202050,34763300,-118195220,449141836,-1551580888,
%U 5924217936,-20734762776,79483257308,-281248448936,1081724803600,-3863302870000,14901311070000,-53644719852000
%N Moment sequence of tr(A^2) in USp(4).
%C If A is a random matrix in the compact group USp(4) (4 X 4 complex matrices which are unitary and symplectic), then a(n)=E[(tr(A^2)^n] is the n-th moment of the trace of A^2. See A138351 for central moments.
%H Kiran S. Kedlaya and Andrew V. Sutherland, <a href="http://hdl.handle.net/1721.1/64701">Hyperelliptic curves, L-polynomials and random matrices</a>, Massachusetts Institute of Technology. Dept. of Mathematics.
%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]
%F a(n)=(1/2)Integral_{x=0..Pi,y=0..Pi}(2cos(2x)+2cos(2y))^n(2cos(x)-2cos(y))^2(2/Pi*sin^2(x))(2/Pi*sin^2(y))dxdy. a(n)=A126120(n)A138364(n+1)-A138364(n)A126120(n+1)
%F Conjectured e.g.f. BesselI[1,2x](BesselI[0,2x]-BesselI[1,2x])/x. - Benjamin Phillabaum, Feb 25 2011
%e a(5) = -50 because E[(tr(A^2))^5] = -50 for a random matrix A in USp(4).
%e a(5) = A126120(5)A138364(6)-A138364(5)A126120(6) = 0*0-10*5 = -50
%t a[n_] := 1/2*Binomial[2*Floor[n/2]+1, Floor[n/2]+1]*CatalanNumber[1/2*(n+Mod[n, 2])]*(Mod[n, 2]+2); Table[a[n], {n, 0, 27}] (* _Jean-François Alcover_, Mar 13 2014 *)
%Y A signed version of A005558, which is the main entry for this sequence.
%Y Cf. A138349, A138351.
%K sign
%O 0,3
%A _Andrew V. Sutherland_, Mar 16 2008
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