A138549 -id:A138549 - cp's OEIS frontend

A138550 Central moment sequence of t^2 coefficient in det(tI-A) for random matrix A in USp(6).

1, 0, 1, 1, 5, 16, 75, 366, 2016, 11936, 75678, 507575, 3575693, 26289408, 200709665, 1584482382, 12888498820, 107698656192, 922140333952, 8072379904752, 72108967554160, 656190909218560, 6074106708205200, 57118680813847840

Offset: 0

Andrew V. Sutherland, Mar 24 2008

nonn

Let the random variable X be the coefficient of t^2 in the characteristic polynomial det(tI-A) of a random matrix in USp(6) (6x6 complex matrices that are unitary and symplectic). Then a(n) = E[(X-1)^n] is the n-th central moment of X since E[X]=1 (see A138549).

Dimension of space of invariant tensors in second fundamental representation of Sp(6). - Bruce Westbury, Dec 05 2014

			a(4) = 5 because E[(X-1)^4] = 5 for X the t^2 coeff of det(tI-A) in USp(6).

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

Cf. A138540, A138549, A251591.

LiE
```
p_tensor(n,[0,1,0],C3)|[0,0,0]
```

a(n) = Sum_{k=0..n} binomial(n,k)(-1)^{n-k}*A138549(k).

A138551 Moment sequence of t^3 coefficient in det(tI-A) for random matrix A in USp(6).

Original entry on oeis.org

1, 0, 2, 0, 23, 0, 684, 0, 34760, 0, 2493096, 0, 228253267, 0, 25091028820, 0, 3179942075960, 0, 451649016238160, 0, 70421753109861592, 0, 11869050034269797984, 0, 2136758627313217104448, 0

Offset: 0

Andrew V. Sutherland, Mar 24 2008

nonn

Let the random variable X be the coefficient of t^3 in the characteristic polynomial det(tI-A) of a random matrix in USp(6) (6x6 complex matrices that are unitary and symplectic). Then a(n) = E[X^n].

Let L_p(T) be the L-polynomial (numerator of the zeta function) of a genus 3 curve C. Under a generalized Sato-Tate conjecture, for almost all C, a(n) is the n-th moment of the coefficient of t^3 in L_p(t/sqrt(p)), as p varies.

			a(4) = 23 because E[X^4] = 23 for X the t^3 coeff of det(tI-A) in USp(6).

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

Cf. A138540, A138549.

See Prop. 12 of Kedlaya-Sutherland.

cp's OEIS Frontend

A138550 Central moment sequence of t^2 coefficient in det(tI-A) for random matrix A in USp(6).

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