|
|
A138540
|
|
Moment sequence of tr(A) in USp(6).
|
|
14
|
|
|
1, 0, 1, 0, 3, 0, 15, 0, 104, 0, 909, 0, 9449, 0, 112398, 0, 1489410, 0, 21562086, 0, 336086022, 0, 5577242292, 0, 97671172836, 0, 1792348213025, 0, 34268124834495, 0, 679376016769260, 0, 13911118850603610, 0, 293220749128031010, 0
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,5
|
|
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))^n] is the n-th moment of the trace of A.
The multiplicity of the trivial representation in the n-th tensor power of the standard representation of USp(6).
Number of returning walks of length n on a cubic lattice remaining in the chamber x >= y >= z >= 0.
Under a generalized Sato-Tate conjecture, this is the moment sequence of the distribution of unitarized Frobenius traces a_p/sqrt(p) (as p varies), for almost all genus 3 curves.
For genus g the mgf is A(z) = det[F_{i+j-2}(z)], 1<=i,j<=g, where F_m(z) = Sum_j binomial(m,j)(I_{2j-m}(2z)-I_{2j-m+2}) and I_k(z) is the hyperbolic Bessel function (of the first kind) of order k.
Dimension of space of invariant tensors in n-th tensor power of natural representation of Sp(6). - Bruce Westbury, Dec 05 2014
|
|
LINKS
|
|
|
FORMULA
|
mgf: A(z) = det[F_{i+j-2}(z)], 1<=i,j<=3, where F_m(z) = Sum_j binomial(m,j)(I_{2j-m}(2z)-I_{2j-m+2}(2z)) and I_k(z) is the hyperbolic Bessel function (of the first kind) of order k.
|
|
EXAMPLE
|
a(4)=3 because E[(tr(A)^4] = 3 for a random matrix A in USp(6).
|
|
MATHEMATICA
|
F[m_][z_] := Sum[Binomial[m, j] (BesselI[2j-m, 2z] - BesselI[2j-m+2, 2z]), {j, 0, m}];
A[z_] := Det[Table[F[i+j-2][z], {i, 1, 3}, {j, 1, 3}]];
a[n_] := a[n] = Derivative[n][A][0];
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|