A138349 Moment sequence of tr(A) in USp(4).

1, 0, 1, 0, 3, 0, 14, 0, 84, 0, 594, 0, 4719, 0, 40898, 0, 379236, 0, 3711916, 0, 37975756, 0, 403127256, 0, 4415203280, 0, 49671036900, 0, 571947380775, 0, 6721316278650, 0, 80419959684900, 0, 977737404590100, 0, 12058761323277900, 0

Offset

0,5

Comments

An aerated version of A005700, which is the main entry for this sequence.

If A is a random matrix in the compact group USp(4) (4 X 4 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(4).

Number of returning NESW walks of length n on a 2-d integer lattice remaining in the chamber x>=y>=0, same as A005700(n/2) for n even.

Under a generalized Sato-Tate conjecture, this is the moment sequence of the distribution of scaled Frobenius traces a_p/sqrt(p) (as p varies), for almost all genus 2 curves. - Andrew V. Sutherland, Mar 16 2008

Links

Table of n, a(n) for n=0..37.

Bostan, Alin ; Chyzak, Frédéric; van Hoeij, Mark; Kauers, Manuel; Pech, Lucien Hypergeometric expressions for generating functions of walks with small steps in the quarter plane. Eur. J. Comb. 61, 242-275 (2017), Table 3

David J. Grabiner and Peter Magyar, Random walks in Weyl chambers and the decomposition of tensor powers, Journal of Algebraic Combinatorics, vol. 2 (1993), no. 3, pp 239-260.

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

Formula

a(n) = (1/2)Integral_{x=0..Pi,y=0..Pi}(2cos(x)+2cos(y))^n(2cos(x)-2cos(y))^2(2/Pi*sin^2(x))(2/Pi*sin^2(y))dxdy.

a(n) = A126120(n)*A126120(n+4)-A126120(n+2)^2.

a(2n) = A005700(n) = A000108(n)*A000108(n+2)-A000108(n+1)^2, a(2n+1)=0.

Example

a(4)=3 because E[(tr(A)^4] = 3 for a random matrix A in USp(4).
a(4)=3 because A126120(4)A126120(8)-A126120(6)^2 = 2*14-5*5 = 3.
a(4)=3 because EEWW, EWEW and ENSW are the returning walks on Z^2 with x>=y>=0.

Crossrefs

Cf. A005700, A126120, A000108.

Sequence in context: A181905 A350826 A008403 * A369918 A323689 A321413

Adjacent sequences: A138346 A138347 A138348 * A138350 A138351 A138352

Keyword

easy,nonn

Author

Andrew V. Sutherland, Mar 16 2008

Status

approved