0,5

If A is a random matrix in the compact group USp(6) (6x6 complex matrices which are unitary and symplectic), then a(n)=E[(tr(A))^n] is the nth moment of the trace of A.

The multiplicity of the trivial representation in the nth tensor power of the standard representation of USp(6).

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

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.

Kiran S. Kedlaya and Andrew V. Sutherland, "Hyperelliptic curves, L-polynomials and random matrices", preprint, 2008.

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.

Nicholas M. Katz and Peter Sarnak, "Random Matrices, Frobenius Eigenvalues and Monodromy", AMS, 1999.

Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices.

mgf is A(z) = det[F_{i+j-2}(z)], 1<=i,j<=g, where F_m(z) = Sum_j Binom(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.

mgf: A(z) = det[F_{i+j-2}(z)], 1<=i,j<=3, where F_m(z) = Sum_j Binom(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.

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

Cf. 138349.

Sequence in context: A135399 A065121 A167339 this_sequence A123023 A130637 A054882

Adjacent sequences: A138537 A138538 A138539 this_sequence A138541 A138542 A138543

nonn

Andrew V. Sutherland (drew(AT)math.mit.edu), Mar 24 2008, Apr 01 2008