This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A277660 #50 Aug 27 2025 06:31:30
%S A277660 0,0,2,30,310,2730,21980,167076,1220100,8650620,59958030,408172050,
%T A277660 2738441706,18151701750,119100934680,774719545320,5001728701800,
%U A277660 32081745977496,204596905143930,1298154208907430,8199305968563710,51576591659861730,323239814342259892,2019025558874685900
%N A277660 2nd-order coefficients of the 1/N-expansion of traces of negative powers of complex Wishart matrices with parameter c=2.
%C A277660 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.)
%H A277660 G. C. Greubel, <a href="/A277660/b277660.txt">Table of n, a(n) for n = 0..1000</a>
%H A277660 F. D. Cunden, F. Mezzadri, N. Simm and P. Vivo, <a href="http://dx.doi.org/10.1063/1.4966642">Large-N expansion for the time-delay matrix of ballistic chaotic cavities</a>, J. Math. Phys. 57, 111901 (2016).
%H A277660 J. Kuipers, M. Sieber and D. Savin, <a href="http://dx.doi.org/10.1088/1367-2630/16/12/123018">Efficient semiclassical approach for time delays</a>, New J. Phys. 16 (2014), 123018.
%F A277660 G.f.: (2*x^2)/(x^2-6*x+1)^(5/2).
%F A277660 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
%F A277660 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
%F A277660 From _Peter Bala_, Sep 20 2024: (Start)
%F A277660 a(n) = (1/6) * Sum_{k = 0..n} k*(k - 1)*binomial(n, k)*binomial(n+k, k).
%F A277660 a(n) = (1/12)*n*(n + 1)*(n - 1)*(n + 2)*hypergeom([n+3, -n+2], [3], -1).
%F A277660 a(n) = (2/3) * d^2/dx^2(Legendre_P(n, x)) at x = 3.
%F A277660 a(n) = (1/12)*n*(n + 1)*A001850(n) - (1/2)*A002695(n).
%F A277660 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)
%p A277660 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:
%p A277660 seq(a(n), n = 1..25); # _Peter Bala_, Sep 28 2024
%t A277660 a[n_] := If[n<2, 0, 2 GegenbauerC[n-2, 5/2, 3]]; a /@ Range[0, 20] (* _Andrey Zabolotskiy_, Oct 27 2016 *)
%t A277660 CoefficientList[Series[(2 x^2) / (x^2 - 6 x + 1)^(5/2), {x, 0, 25}], x] (* _Vincenzo Librandi_, Oct 30 2016 *)
%o A277660 (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
%Y A277660 Cf. A001850, A002695, A277661, A277662, A277663, A277664, A277665.
%K A277660 nonn,easy,changed
%O A277660 0,3
%A A277660 _Fabio Deelan Cunden_, Oct 26 2016
%E A277660 a(9)-a(22) from _Andrey Zabolotskiy_, Oct 26 2016
%E A277660 a(23) from _Fabio Deelan Cunden_, Oct 29 2016