A138350 Moment sequence of tr(A^2) in USp(4).

1, -1, 3, -6, 20, -50, 175, -490, 1764, -5292, 19404, -60984, 226512, -736164, 2760615, -9202050, 34763300, -118195220, 449141836, -1551580888, 5924217936, -20734762776, 79483257308, -281248448936, 1081724803600, -3863302870000, 14901311070000, -53644719852000

Offset: 0

Author

Andrew V. Sutherland, Mar 16 2008

Keywords

• sign

Comments

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^2))^n] is the n-th moment of the trace of A^2. See A138351 for central moments.

Examples

			a(5) = -50 because E[(tr(A^2))^5] = -50 for a random matrix A in USp(4).
a(5) = A126120(5)*A138364(6)-A138364(5)*A126120(6) = 0*0-10*5 = -50.
		

Links

• Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, Massachusetts Institute of Technology. Dept. of Mathematics.
• Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT]

Crossrefs

A signed version of A005558, which is the main entry for this sequence.

Cf. A138349, A138351.

Programs

Mathematica

a[n_] := 1/2*Binomial[2*Floor[n/2]+1, Floor[n/2]+1]*CatalanNumber[1/2*(n+Mod[n, 2])]*(Mod[n, 2]+2); Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Mar 13 2014 *)

Formula

a(n)=(1/2)Integral_{x=0..Pi,y=0..Pi}(2cos(2x)+2cos(2y))^n(2cos(x)-2cos(y))^2(2/Pi*sin^2(x))(2/Pi*sin^2(y))dxdy. a(n)=A126120(n)A138364(n+1)-A138364(n)A126120(n+1)

Conjectured e.g.f. BesselI[1,2x](BesselI[0,2x]-BesselI[1,2x])/x. - Benjamin Phillabaum, Feb 25 2011