login
A277660
2nd-order coefficients of the 1/N-expansion of traces of negative powers of complex Wishart matrices with parameter c=2.
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
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
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)
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 *)
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
KEYWORD
nonn,easy
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