A272865 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.

4, 24, 160, 132, 936, 5700, 720, 5312, 33264, 198144, 3940, 29880, 190980, 1155600, 6823620, 21672, 167712, 1088856, 6670656, 39786120, 233908896, 119812, 941640, 6189540, 38300976, 230340740, 1363667256, 7997325700

Offset

1,1

Comments

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

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

References

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

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).

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).

Links

Table of n, a(n) for n=1..28.

F. D. Cunden, Statistical distribution of the Wigner-Smith time-delay matrix moments for chaotic cavities, arXiv:1412.2172 [cond-mat.mes-hall], 2014-2015.

F. D. Cunden, F. Mezzadri, N. Simm and P. Vivo, Correlators for the Wigner-Smith time-delay matrix of chaotic cavities, arXiv:1601.06690 [math-ph], 2016.

F. D. Cunden, F. Mezzadri, N. O'Connell and N. Simm, Moments of Random Matrices and Hypergeometric Orthogonal Polynomials, arXiv:1805.08760 [math-ph], 2018.

Formula

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

T(n,1)/4 = A050151(n) for n>=1. - Peter Luschny, May 08 2016

Example

Triangle starts:
4;
24,   160;
132,  936,   5700;
720,  5312,  33264,  198144;
3940, 29880, 190980, 1155600, 6823620;

Maple

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

Mathematica

Clear["Global`*"]; (*Wigner-Smith Covariance*)
P[k_] := Sum[Binomial[k - 1, j] Binomial[k + j, j], {j, 0, k - 1}]
Q[k_] := Sum[Binomial[k, j + 1] Binomial[k + j, j], {j, 0, k - 1}]
a[k1_, k2_] := 4 (k1 k2)/(k1 + k2) (P[k1] Q[k2] + P[k2] Q[k1])
L = 10; Table[a[k, l], {k, 1, L}, {l, 1, k}]

Crossrefs

Cf. A002002, A047781, A050151.

Sequence in context: A210474 A344735 A117337 * A084130 A059304 A069722

Adjacent sequences: A272862 A272863 A272864 * A272866 A272867 A272868

Keyword

nonn,tabl

Author

Fabio Deelan Cunden, May 08 2016

Status

approved