|
| |
|
|
A138544
|
|
Moment sequence of tr(A^4) in USp(6).
|
|
1
| |
|
|
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
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,3
|
|
|
COMMENTS
| If A is a random matrix in the compact group USp(6) (6x6 complex
matrices which are unitary and symplectic), then a(n)=E[(tr(A^4))^n] is the nth
moment of the trace of A^4. See A138545 for central moments.
|
|
|
REFERENCES
| Kiran S. Kedlaya and Andrew V. Sutherland, "Hyperelliptic curves, L-polynomials and random matrices", preprint, 2008.
|
|
|
FORMULA
| mgf is A(z) = det[F_{i+j-2}(z)], 1<=i,j<=3, where F_m(z) = Sum_j Binom(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.
|
|
|
EXAMPLE
| a(3) = -9 because E[(tr(A^4))^3] = -9 for a random matrix A in USp(6).
|
|
|
CROSSREFS
| Cf. A138540, A138545.
Sequence in context: A149167 A149168 A149169 * A093149 A048054 A149170
Adjacent sequences: A138541 A138542 A138543 * A138545 A138546 A138547
|
|
|
KEYWORD
| sign
|
|
|
AUTHOR
| Andrew V. Sutherland (drew(AT)math.mit.edu), Mar 24 2008
|
| |
|
|
