A277660 2nd-order coefficients of the 1/N-expansion of traces of negative powers of complex Wishart matrices with parameter c=2.

0, 0, 2, 30, 310, 2730, 21980, 167076, 1220100, 8650620, 59958030, 408172050, 2738441706, 18151701750, 119100934680, 774719545320, 5001728701800, 32081745977496, 204596905143930, 1298154208907430, 8199305968563710, 51576591659861730, 323239814342259892, 2019025558874685900

Offset

0,3

Comments

These numbers provide the 2nd order of the 1/N-expansion of traces of powers of a random time-delay matrix without time-reversal symmetry. (The 0th order is instead 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.

Formula

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

a(n) = 2*C_(n-2)^(5/2)(3) for n >= 2, where C_n^(m)(x) is the Gegenbauer polynomial. - Andrey Zabolotskiy, Oct 26 2016

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

Mathematica

a[n_] := If[n<2, 0, 2 GegenbauerC[n-2, 5/2, 3]]; a /@ Range[0, 20] (* Andrey Zabolotskiy, Oct 27 2016 *)
CoefficientList[Series[(2 x^2) / (x^2 - 6 x + 1)^(5/2), {x, 0, 25}], x] (* Vincenzo Librandi, Oct 30 2016 *)

Prog

(PARI) x='x+O('x^50); concat([0, 0], Vec((2*x^2)/(x^2-6*x+1)^(5/2))) \\ G. C. Greubel, Jun 05 2017

Crossrefs

Cf. A277661, A277662, A277663, A277664, A277665.

Sequence in context: A036351 A189770 A245020 * A089433 A152277 A230610

Adjacent sequences: A277657 A277658 A277659 * A277661 A277662 A277663

Keyword

nonn

Author

Fabio Deelan Cunden, Oct 26 2016

Extensions

a(9)-a(22) from Andrey Zabolotskiy, Oct 26 2016

a(23) from Fabio Deelan Cunden, Oct 29 2016

Status

approved