A138546 Moment sequence of tr(A^5) in USp(6).

1, 0, 4, 0, 42, 0, 660, 0, 12810, 0, 281736, 0, 6727644, 0, 170316432, 0, 4504487130, 0, 123255492360, 0, 3465702008340, 0, 99645553785960, 0, 2918768920720380, 0, 86852063374902000, 0, 2619552500788984200, 0, 79939673971478231760, 0

Offset

0,3

Comments

If A is a random matrix in the compact group USp(6) (6 X 6 complex matrices that are unitary and symplectic), then a(n) = E[(tr(A^5))^n] is the n-th moment of the trace of A^5.

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)/5}(z)-B_{(2j-m+2)/5}(z)) and B_v(z)=0 for non-integer v and otherwise B_v(z)=I_v(2z) with I_v(z) the hyperbolic Bessel function (of the first kind) of order v.

Example

a(4) = 42 because E[(tr(A^5))^4] = 42 for a random matrix A in USp(6).

Crossrefs

Cf. A138540.

Sequence in context: A174083 A123936 A271834 * A019217 A221757 A189424

Adjacent sequences: A138543 A138544 A138545 * A138547 A138548 A138549

Keyword

nonn

Author

Andrew V. Sutherland, Mar 24 2008

Status

approved