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

Author

Andrew V. Sutherland, Mar 16 2008

Keywords

• easy
• nonn

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

Examples

			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.
		

Links

• 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.

Crossrefs

Cf. A005700, A126120, A000108.

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.