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^6))^n] is the n-th moment of the trace of A^6. See A138547 for central moments.
LINKS
Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.
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)/6}(z)-B_{(2j-m+2)/6}(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(3) = -15 because E[(tr(A^6))^3] = -15 for a random matrix A in USp(6).
CROSSREFS
KEYWORD
sign
AUTHOR
Andrew V. Sutherland, Mar 24 2008
STATUS
approved