A138547 Moment sequence of tr(A^6) in USp(6).

1, -1, 6, -15, 90, -310, 1860, -7455, 44730, -195426, 1172556, -5416026, 32496156, -156061620, 936369720, -4628393055, 27770358330, -140348412490, 842090474940, -4331544836190, 25989269017140, -135614951248140, 813689707488840, -4296741195214650, 25780447171287900

Offset: 0

Author

Andrew V. Sutherland, Mar 24 2008

Keywords

• sign

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.

Examples

			a(3) = -15 because E[(tr(A^6))^3] = -15 for a random matrix A in USp(6).
		

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.

Crossrefs

Cf. A138540, A138547.

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.