A138540 -id:A138540 - cp's OEIS frontend

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

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.

A174516 Partial sums of A002896.

Original entry on oeis.org

1, 7, 97, 1957, 46687, 1219243, 33715399, 970085119, 28740443449, 870830918389, 26860099935529, 840549807424369, 26620996978712269, 851664885506669269, 27482469263443730269, 893460843597349019629, 29235859228655427097639

Offset: 0

Jonathan Vos Post, Mar 20 2010

nonn

			a(4) = 1 + 6 + 90 + 1860 + 44730 = 46687.

Cf. A002896, A049020, A049037, A084261, A138540, A140476.

b[n_] := b[n] = (* A002896 *) Binomial[2*n, n]*HypergeometricPFQ[{1/2, -n, -n}, {1, 1}, 4]; a[n_] := Sum[b[k], {k, 0, n}]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Dec 20 2011 *)

a(n) = Sum_{i=0..n} A002896(i).
G.f.: g/(1-x) where g is the o.g.f. of A002896. - Mark van Hoeij, Nov 12 2011
a(n) ~ 2^(2*n) * 3^(2*n + 7/2) / (35 * Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Feb 17 2024

A251591 Dimension of space of invariant tensors in 2n-th tensor power of the third fundamental representation of Sp(6).

Original entry on oeis.org

1, 1, 4, 35, 560, 14973, 589743, 30078048, 1824041570, 125400975830, 9507019477382, 78070828079199, 68560287232877740, 6376178095301876005, 623169409884847073730, 636070059202675270255520, 6745818886029778590765570, 740194253157571009569356970

Offset: 0

Bruce Westbury, Dec 05 2014

Cf. A138540, A138550.

LiE
```
p_tensor(2*n,[0,0,1],C3)|[0,0,0]
```

A138548 Central moment sequence of tr(A^6) in USp(6).

Original entry on oeis.org

1, 0, 5, 1, 63, 46, 1135, 1800, 25431, 66232, 666387, 2397605, 19650565, 87187842, 633498229, 3214996309, 21829972815, 120665223560, 790528831099, 4613644505799, 29715748525937, 179604102525370, 1149406514424945

Offset: 0

Andrew V. Sutherland, Mar 24 2008

nonn

If A is a random matrix in the compact group USp(6) (6x6 complex matrices that are unitary and symplectic), then a(n)=E[(tr(A^6)+1)^n] is the n-th central moment of the trace of A^6, since E[tr(A^6)] = -1 (see A138546).

			a(5) = 46 because E[(tr(A^6)+1)^5] = 46 for a random matrix 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, A138546.

mgf is A(z)=e^zF(z) where F(z) is the mgf of A138546.

cp's OEIS Frontend

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

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A174516 Partial sums of A002896.

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A251591 Dimension of space of invariant tensors in 2n-th tensor power of the third fundamental representation of Sp(6).

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LiE

A138548 Central moment sequence of tr(A^6) in USp(6).

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