A138541 Moment sequence of tr(A^2) in USp(6).

1, -1, 3, -7, 24, -75, 285, -1036, 4242, -16926, 73206, -311256, 1403028, -6247527, 29082339, -134138290, 640672890, -3038045010, 14818136190, -71858704710, 356665411440, -1761879027090, 8874875097270, -44526516209280, 227135946200940, -1154738374364100, 5955171596514900

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^2))^n] is the n-th moment of the trace of A^2. See A138542 for central moments.

Links

Table of n, a(n) for n=0..26.

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<=g, where F_m(z) = Sum_j binomial(m,j)(B_{(2j-m)/2}(z)-B_{(2j-m+2)/2}(z)) and B_v(z)=0 for non-integer k 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) = 24 because E[(tr(A^2))^4] = 24 for a random matrix A in USp(6).

Crossrefs

Cf. A138540, A138542.

Sequence in context: A003449 A258308 A148719 * A290750 A148720 A225826

Adjacent sequences: A138538 A138539 A138540 * A138542 A138543 A138544

Keyword

sign

Author

Andrew V. Sutherland, Mar 24 2008

Status

approved