A272865 - OEIS

%I #18 Jul 20 2019 12:27:14

%S 4,24,160,132,936,5700,720,5312,33264,198144,3940,29880,190980,

%T 1155600,6823620,21672,167712,1088856,6670656,39786120,233908896,

%U 119812,941640,6189540,38300976,230340740,1363667256,7997325700

%N Triangle read by rows, T(n,k) are covariances of inverse power traces of complex Wishart matrices with parameter c=2, for n>=1 and 1<=k<=n.

%C These numbers provide the covariances of power traces of the time-delay matrix when the scattering matrix belongs to the Dyson ensembles.

%C Relation with A047781 and A002002. See eq. (60) and (61) in Cunden et al., J. Phys. A: Math. Theor. 49, 18LT01 (2016).

%D F. D. Cunden, "Statistical distribution of the Wigner-Smith time-delay matrix moments for chaotic cavities", Phys. Rev. E 91, 060102(R) (2015).

%D F. D. Cunden, F. Mezzadri, N. Simm and P. Vivo, "Correlators for the Wigner-Smith time-delay matrix of chaotic cavities", J. Phys. A: Math. Theor. 49, 18LT01 (2016).

%D F. D. Cunden, F. Mezzadri, N. O'Connell and N. Simm, "Moments of Random Matrices and Hypergeometric Orthogonal Polynomials", Commun. Math. Phys. 369, 1091-1145 (2019).

%H F. D. Cunden, <a href="http://arxiv.org/abs/1412.2172">Statistical distribution of the Wigner-Smith time-delay matrix moments for chaotic cavities</a>, arXiv:1412.2172 [cond-mat.mes-hall], 2014-2015.

%H F. D. Cunden, F. Mezzadri, N. Simm and P. Vivo, <a href="http://arxiv.org/abs/1601.06690">Correlators for the Wigner-Smith time-delay matrix of chaotic cavities</a>, arXiv:1601.06690 [math-ph], 2016.

%H F. D. Cunden, F. Mezzadri, N. O'Connell and N. Simm, <a href="https://arxiv.org/abs/1805.08760">Moments of Random Matrices and Hypergeometric Orthogonal Polynomials</a>, arXiv:1805.08760 [math-ph], 2018.

%F G.f.: ((x*y)/(x-y)^2)*((x*y-3(x+y)+1)/(sqrt(x^2-6x+1)*sqrt(y^2-6y+1))-1).

%F T(n,1)/4 = A050151(n) for n>=1. - _Peter Luschny_, May 08 2016

%e Triangle starts:

%e 4;

%e 24,   160;

%e 132,  936,   5700;

%e 720,  5312,  33264,  198144;

%e 3940, 29880, 190980, 1155600, 6823620;

%p P := (n,k) -> simplify(n*hypergeom([1-k,k+1],[1],-1)*hypergeom([1-n,n+1],[2],-1)): seq(seq(4*(n*k)*(P(n,k)+P(k,n))/(n+k),k=1..n),n=1..7); # _Peter Luschny_, May 08 2016

%t Clear["Global`*"];(*Wigner-Smith Covariance*)

%t P[k_] := Sum[Binomial[k - 1, j] Binomial[k + j, j], {j, 0, k - 1}]

%t Q[k_] := Sum[Binomial[k, j + 1] Binomial[k + j, j], {j, 0, k - 1}]

%t a[k1_, k2_] := 4 (k1 k2)/(k1 + k2) (P[k1] Q[k2] + P[k2] Q[k1])

%t L = 10; Table[a[k, l], {k, 1, L}, {l, 1, k}]

%Y Cf. A002002, A047781, A050151.

%K nonn,tabl

%O 1,1

%A _Fabio Deelan Cunden_, May 08 2016