

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
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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 nth moment of the trace of A.
The multiplicity of the trivial representation in the nth 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 SatoTate 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+j2}(z)], 1<=i,j<=g, where F_m(z) = Sum_j binomial(m,j)(I_{2jm}(2z)I_{2jm+2}) and I_k(z) is the hyperbolic Bessel function (of the first kind) of order k.
Dimension of space of invariant tensors in nth tensor power of natural representation of Sp(6).  Bruce Westbury, Dec 05 2014


LINKS



FORMULA

mgf: A(z) = det[F_{i+j2}(z)], 1<=i,j<=3, where F_m(z) = Sum_j binomial(m,j)(I_{2jm}(2z)I_{2jm+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[2jm, 2z]  BesselI[2jm+2, 2z]), {j, 0, m}];
A[z_] := Det[Table[F[i+j2][z], {i, 1, 3}, {j, 1, 3}]];
a[n_] := a[n] = Derivative[n][A][0];


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



