A277662 2nd-order coefficients of the 1/N-expansion of traces of negative powers of real Wishart matrices with parameter c=2.

0, 0, 6, 102, 1142, 10650, 89576, 705012, 5297924, 38478492, 272262050, 1887071274, 12862479402, 86468603910, 574580180020, 3780504491400, 24663229376872, 159709443132888, 1027505285362590, 6572573611318158, 41827041105943870, 264959521695360786, 1671472578046156512, 10504743400858155708

Offset: 0

Author

Fabio Deelan Cunden, Oct 26 2016

Keywords

• nonn

Comments

These numbers provide the 2nd 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).
• J. Kuipers, M. Sieber and D. Savin, Efficient semiclassical approach for time delays, New J. Phys. 16 (2014), 123018.

Crossrefs

Cf. A006318 (0th order), A277661 (1st order), A277663 (3rd order), A277664 (4th order), A277665 (5th order).

Programs

Mathematica

CoefficientList[Series[(x^2 - 3 x)/((x^2 - 6 x + 1)^2) + (3 x^3 - 4 x^2 + 3 x)/((x^2 - 6 x + 1)^(5/2)), {x, 0, 23}], x] (* Michael De Vlieger, Oct 26 2016 *)

Formula

G.f.: (x^2-3*x)/((x^2-6*x+1)^2)+(3*x^3-4*x^2+3 x)/((x^2-6*x+1)^(5/2)).

a(n) ~ 7*(3*sqrt(2)+4)^(5/2) * n^(3/2) * (1+sqrt(2))^(2*n-4) / (3*2^(9/2)*sqrt(Pi)) * (1 - (3*sqrt((2+3/sqrt(2))*Pi))/(7*sqrt(n))). - Vaclav Kotesovec, Oct 27 2016

Extensions

More terms from Michael De Vlieger, Oct 26 2016