This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A138349 #13 Jan 31 2025 01:45:05
%S A138349 1,0,1,0,3,0,14,0,84,0,594,0,4719,0,40898,0,379236,0,3711916,0,
%T A138349 37975756,0,403127256,0,4415203280,0,49671036900,0,571947380775,0,
%U A138349 6721316278650,0,80419959684900,0,977737404590100,0,12058761323277900,0
%N A138349 Moment sequence of tr(A) in USp(4).
%C A138349 An aerated version of A005700, which is the main entry for this sequence.
%C A138349 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.
%C A138349 The multiplicity of the trivial representation in the n-th tensor power of the standard representation of USp(4).
%C A138349 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.
%C A138349 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
%H A138349 Bostan, Alin ; Chyzak, Frédéric; van Hoeij, Mark; Kauers, Manuel; Pech, Lucien <a href="https://doi.org/10.1016/j.ejc.2016.10.010">Hypergeometric expressions for generating functions of walks with small steps in the quarter plane.</a> Eur. J. Comb. 61, 242-275 (2017), Table 3
%H A138349 David J. Grabiner and Peter Magyar, <a href="http://dx.doi.org/10.1023/A:1022499531492">Random walks in Weyl chambers and the decomposition of tensor powers</a>, Journal of Algebraic Combinatorics, vol. 2 (1993), no. 3, pp 239-260.
%H A138349 Kiran S. Kedlaya and Andrew V. Sutherland, <a href="http://arXiv.org/abs/0803.4462">Hyperelliptic curves, L-polynomials and random matrices</a>, arXiv:0803.4462 [math.NT], 2008-2010.
%F A138349 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.
%F A138349 a(n) = A126120(n)*A126120(n+4)-A126120(n+2)^2.
%F A138349 a(2n) = A005700(n) = A000108(n)*A000108(n+2)-A000108(n+1)^2, a(2n+1)=0.
%e A138349 a(4)=3 because E[(tr(A))^4] = 3 for a random matrix A in USp(4).
%e A138349 a(4)=3 because A126120(4)A126120(8)-A126120(6)^2 = 2*14-5*5 = 3.
%e A138349 a(4)=3 because EEWW, EWEW and ENSW are the returning walks on Z^2 with x>=y>=0.
%Y A138349 Cf. A005700, A126120, A000108.
%K A138349 easy,nonn
%O A138349 0,5
%A A138349 _Andrew V. Sutherland_, Mar 16 2008