A277663 3rd-order coefficients of the 1/N-expansion of traces of negative powers of real Wishart matrices with parameter c=2.

0, 0, 10, 378, 7048, 96000, 1092460, 11060700, 103150528, 905077728, 7576640950, 61098854454, 477942694136, 3645484792560, 27220292840440, 199588002587160, 1440630859132416, 10256896070590464, 72150109176698562, 502120765832371602, 3461203073248719400, 23654601049848668256

Offset: 0

Author

Fabio Deelan Cunden, Oct 26 2016

Keywords

• nonn

Comments

These numbers provide the 3rd order of the 1/N-expansion of traces of powers of a random time-delay matrix in presence of time-reversal symmetry. (The 0th order is given by the Large Schröder numbers A006318.)

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• F. D. Cunden, F. Mezzadri, N. Simm and P. Vivo, Large-N expansion for the time-delay matrix of ballistic chaotic cavities, J. Math. Phys. 57, 111901 (2016). E-print arXiv:1607.00250.
• J. Kuipers, M. Sieber and D. Savin, Efficient semiclassical approach for time delays, New J. Phys. 16, 123018 (2014), arXiv:1409.1532 [nlin.CD], 2014.

Crossrefs

Cf. A006318 (0th order), A277661 (1st order), A277662 (2nd order), A277664 (4th order), A277665 (5th order).

Programs

Mathematica

CoefficientList[Series[-(2 x) (2 x^3 - 9 x^2 + 19 x + 3) / ((x^2 - 6 x + 1)^(7/2)) - (2 x) (6 x^4 - 5 x^3 + 9 x^2 - 15 x - 3) / ((x^2 - 6 x + 1)^4), {x, 0, 25}], x] (* Vincenzo Librandi, Nov 07 2016 *)

Formula

G.f.: -(2*z)*(2*z^3-9*z^2+19*z+3)/(y(z)^(7/2))-(2*z)*(6*z^4-5*z^3+9*z^2-15*z-3)/(y(z)^4) where y(z)=z^2-6*z+1.

a(n) ~ (17*sqrt(2)/24-1) * n^3 * (1+sqrt(2))^(2*n+6) * (1 - (7*sqrt((8+6*sqrt(2)) / Pi))/(8*sqrt(n))). - Vaclav Kotesovec, Oct 27 2016