A138544 Moment sequence of tr(A^4) in USp(6).

1, -1, 4, -9, 42, -130, 660, -2415, 12810, -51786, 281736, -1216446, 6727644, -30440124, 170316432, -798126615, 4504487130, -21692469370, 123255492360, -606672653730, 3465702008340, -17366224451940, 99645553785960, -506814533253210, 2918768920720380, -15034038412333500

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 which are unitary and symplectic), then a(n) = E[(tr(A^4))^n] is the n-th moment of the trace of A^4. See A138545 for central moments.

Examples

			a(3) = -9 because E[(tr(A^4))^3] = -9 for a random matrix A in USp(6).
		

Links

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

Crossrefs

Cf. A138540, A138545.

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)/4}(z)-B_{(2j-m+2)/4}(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.