%I A138349
%S A138349 1,0,1,0,3,0,14,0,84,0,594,0,4719,0,40898,0,379236,0,3711916,0,37975756,
%T A138349 0,403127256,0,4415203280,0,49671036900,0,571947380775,0,6721316278650,
%U A138349 0,80419959684900,0,977737404590100,0,12058761323277900,0
%N A138349 Moment sequence of tr(A) in USp(4).
%C A138349 An aerated version of A005700, which is the main entry for this sequence.
%C A138349 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))^n] is the 
               nth moment of the trace of A.
%C A138349 The multiplicity of the trivial representation in the nth tensor power 
               of the standard representation of USp(4).
%C A138349 Number of returning NESW walks of length n on a 2-d integer lattice remaining 
               in the chamber x>=y>=0, same as A005700(n/2) for n even.
%C A138349 Under a generalized Sato-Tate conjecture, this is the moment sequence 
               of the distribution of scaled Frobenius traces a_p/sqrt(p) (as p 
               varies), for almost all genus 2 curves. - Andrew V. Sutherland (drew(AT)math.mit.edu), 
               Mar 16 2008
%D A138349 David J. Grabiner and Peter Magyar, "Random walks in Weyl chambers and 
               the decomposition of tensor powers", Journal of Algebraic Combinatorics, 
               vol. 2 (1993), no. 3, pp 239-260.
%D A138349 Kiran S. Kedlaya and Andrew V. Sutherland, "Hyperelliptic curves, L-polynomials 
               and random matrices", preprint, 2008.
%H A138349 Kiran S. Kedlaya and Andrew V. Sutherland, <a href="http://arXiv.org/
               abs/0803.4462">Hyperelliptic curves, L-polynomials and random matrices</
               a>.
%F A138349 a(n)=(1/2)Integral_{x=0..Pi,y=0..Pi}(2cos(x)+2cos(y))^n(2cos(x)-2cos(y))^2(2/
               Pi*sin^2(x))(2/Pi*sin^2(y))dxdy. a(n)=A126120(n)A126120(n+4)-A126120(n+2)^2. 
               a(2n)=A005700(n)=A000108(n)A000108(n+2)-A000108(n+1)^2, a(2n+1)=0.
%e A138349 a(4)=3 because E[(tr(A)^4] = 3 for a random matrix A in USp(4).
%e A138349 a(4)=3 because A126120(4)A126120(8)-A126120(6)^2 = 2*14-5*5 = 3.
%e A138349 a(4)=3 because EEWW, EWEW and ENSW are the returning walks on Z^2 with 
               x>=y>=0.
%Y A138349 Cf. A005700, A126120, A000108.
%Y A138349 Sequence in context: A057374 A058896 A008403 this_sequence A135399 A065121 
               A167339
%Y A138349 Adjacent sequences: A138346 A138347 A138348 this_sequence A138350 A138351 
               A138352
%K A138349 easy,nonn
%O A138349 0,5
%A A138349 Andrew V. Sutherland (drew(AT)math.mit.edu), Mar 16 2008