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

Author

Fabio Deelan Cunden, Oct 26 2016

Keywords

• nonn
• easy
• changed

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.

Crossrefs

Cf. A001850, A002695, A277661, A277662, A277663, A277664, A277665.

Programs

Maple

a := proc(n) option remember; if n = 1 then 0 elif n = 2 then 2 else (3*(2*n - 1)*a(n-1) - (n + 1)*a(n-2))/(n - 2) fi; end:
seq(a(n), n = 1..25); # Peter Bala, Sep 28 2024

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 *)

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

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) ~ (1 + sqrt(2))^(2*n+1) * n^(3/2) / (3*2^(13/4)*sqrt(Pi)). - Vaclav Kotesovec, Oct 27 2016, simplified Aug 27 2025

From Peter Bala, Sep 20 2024: (Start)

a(n) = (1/6) * Sum_{k = 0..n} k*(k - 1)*binomial(n, k)*binomial(n+k, k).

a(n) = (1/12)*n*(n + 1)*(n - 1)*(n + 2)*hypergeom([n+3, -n+2], [3], -1).

a(n) = (2/3) * d^2/dx^2(Legendre_P(n, x)) at x = 3.

a(n) = (1/12)*n*(n + 1)*A001850(n) - (1/2)*A002695(n).

P-recursive: (n - 2)*a(n) = 3*(2*n - 1)*a(n-1) - (n + 1)*a(n-2) with a(1) = 0 and a(2) = 2. (End)

Extensions

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

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