A138543 Moment sequence of tr(A^3) in USp(6).

1, 0, 3, 0, 26, 0, 345, 0, 5754, 0, 110586, 0, 2341548, 0, 53208441, 0, 1276027610, 0, 31930139670, 0, 826963069140, 0, 22035414489270, 0, 601361536493340, 0, 16749316314679500, 0, 474777481850283240, 0, 13665774112508864385, 0

Offset

0,3

Comments

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^3))^n] is the n-th moment of the trace of A^3.

Links

Table of n, a(n) for n=0..31.

Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2008-2010.

Formula

mgf is A(z) = det[F_{i+j-2}(z)], 1<=i,j<=3, where F_m(z) = Sum_j binomial(m,j)(B_{(2j-m)/3}(z)-B_{(2j-m+2)/3}(z)) and B_v(z)=0 for non-integer v and otherwise B_v(z)=I_v(2z) with I_v(z) is the hyperbolic Bessel function (of the first kind) of order v.

Example

a(4) = 26 because E[(tr(A^2))^4] = 26 for a random matrix A in USp(6).

Crossrefs

Cf. A138540.

Sequence in context: A057379 A360063 A009777 * A238104 A143769 A190963

Adjacent sequences: A138540 A138541 A138542 * A138544 A138545 A138546

Keyword

nonn

Author

Andrew V. Sutherland, Mar 24 2008

Status

approved