0,5

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 nth moment of the trace of A.

The multiplicity of the trivial representation in the nth 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 (drew(AT)math.mit.edu), Mar 16 2008

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", preprint, 2008.

Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices.

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.

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.

Cf. A005700, A126120, A000108.

Sequence in context: A057374 A058896 A008403 this_sequence A135399 A065121 A167339

Adjacent sequences: A138346 A138347 A138348 this_sequence A138350 A138351 A138352

easy,nonn

Andrew V. Sutherland (drew(AT)math.mit.edu), Mar 16 2008