%I A138350
%S A138350 1,1,3,6,20,50,175,490,1764,5292,19404,60984,226512,736164,2760615,9202050,
%T A138350 34763300,118195220,449141836,1551580888,5924217936,20734762776,79483257308,
%U A138350 281248448936,1081724803600,3863302870000,14901311070000,53644719852000
%V A138350 1,-1,3,-6,20,-50,175,-490,1764,-5292,19404,-60984,226512,-736164,2760615,
               -9202050,
%W A138350 34763300,-118195220,449141836,-1551580888,5924217936,-20734762776,79483257308,
%X A138350 -281248448936,1081724803600,-3863302870000,14901311070000,-53644719852000
%N A138350 Moment sequence of tr(A^2) in USp(4).
%C A138350 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.
%D A138350 Kiran S. Kedlaya and Andrew V. Sutherland, "Hyperelliptic curves, L-polynomials 
               and random matrices", preprint, 2008.
%H A138350 Kiran S. Kedlaya and Andrew V. Sutherland, <a href="http://arXiv.org/
               abs/0803.4462">Hyperelliptic curves, L-polynomials and random matrices</
               a>.
%F A138350 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)
%e A138350 a(5) = -50 because E[(tr(A^2))^5] = -50 for a random matrix A in USp(4).
%e A138350 a(5) = A126120(5)A138364(6)-A138364(5)A126120(6) = 0*0-10*5 = -50
%Y A138350 A signed version of A005558, which is the main entry for this sequence. 
               Cf. A138349, A138351.
%Y A138350 Sequence in context: A148573 A148574 A005558 this_sequence A148575 A148576 
               A148577
%Y A138350 Adjacent sequences: A138347 A138348 A138349 this_sequence A138351 A138352 
               A138353
%K A138350 sign
%O A138350 0,3
%A A138350 Andrew V. Sutherland (drew(AT)math.mit.edu), Mar 16 2008