A138540 - OEIS

%I #25 Jul 25 2019 03:13:37

%S 1,0,1,0,3,0,15,0,104,0,909,0,9449,0,112398,0,1489410,0,21562086,0,

%T 336086022,0,5577242292,0,97671172836,0,1792348213025,0,

%U 34268124834495,0,679376016769260,0,13911118850603610,0,293220749128031010,0

%N Moment sequence of tr(A) 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))^n] is the n-th moment of the trace of A.

%C The multiplicity of the trivial representation in the n-th tensor power of the standard representation of USp(6).

%C Number of returning walks of length n on a cubic lattice remaining in the chamber x >= y >= z >= 0.

%C Under a generalized Sato-Tate conjecture, this is the moment sequence of the distribution of unitarized Frobenius traces a_p/sqrt(p) (as p varies), for almost all genus 3 curves.

%C For genus g the mgf is A(z) = det[F_{i+j-2}(z)], 1<=i,j<=g, where F_m(z) = Sum_j binomial(m,j)(I_{2j-m}(2z)-I_{2j-m+2}) and I_k(z) is the hyperbolic Bessel function (of the first kind) of order k.

%C Dimension of space of invariant tensors in n-th tensor power of natural representation of Sp(6). - _Bruce Westbury_, Dec 05 2014

%H David J. Grabiner and Peter Magyar, <a href="http://dx.doi.org/10.1023/A:1022499531492">Random walks in Weyl chambers and the decomposition of tensor powers</a>, Journal of Algebraic Combinatorics, vol. 2 (1993), no. 3, pp 239-260.

%H Nicholas M. Katz and Peter Sarnak, <a href="http://bookstore.ams.org/coll-45/">Random Matrices, Frobenius Eigenvalues and Monodromy</a>, AMS, 1999.

%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.

%H G. Lachaud, <a href="http://arxiv.org/abs/1506.06482">On the distribution of the trace in the unitary symplectic group and the distribution of Frobenius</a>, arXiv preprint arXiv:1506.06482 [math.AG], 2015.

%F mgf: A(z) = det[F_{i+j-2}(z)], 1<=i,j<=3, where F_m(z) = Sum_j binomial(m,j)(I_{2j-m}(2z)-I_{2j-m+2}(2z)) and I_k(z) is the hyperbolic Bessel function (of the first kind) of order k.

%e a(4)=3 because E[(tr(A)^4] = 3 for a random matrix A in USp(6).

%t F[m_][z_] := Sum[Binomial[m, j] (BesselI[2j-m, 2z] - BesselI[2j-m+2, 2z]), {j, 0, m}];

%t A[z_] := Det[Table[F[i+j-2][z], {i, 1, 3}, {j, 1, 3}]];

%t a[n_] := a[n] = Derivative[n][A][0];

%t Table[Print[n, " ", a[n]]; a[n], {n, 0, 35}] (* _Jean-François Alcover_, Feb 17 2019 *)

%Y Cf. A138349.

%K nonn

%O 0,5

%A _Andrew V. Sutherland_, Mar 24 2008, Apr 01 2008