A277664 4th-order coefficients of the 1/N-expansion of traces of negative powers of real Wishart matrices with parameter c=2.

0, 0, 22, 1638, 47454, 904530, 13529862, 172576362, 1966038698, 20583987894, 201838423616, 1878183167916, 16744919877108, 144061342087884, 1202594886126228, 9783039293041644, 77823360967288812, 607079393002409364, 4654603707195506610, 35144449267872359562, 261740341786424075106

Offset

0,3

Comments

These numbers provide the 4th 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 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*(36*z^7+20*z^6+24*z^5-219*z^4+216*z^3+163*z^2+6*z))/(y(z)^(11/2)) +(2*(12*z^8-132*z^7+618*z^6-1830*z^5+1840*z^4+720*z^3-134*z^2-6*z))/(y(z)^6), where y(z)= z^2-6*z+1.

a(n) ~ 37 * (3*sqrt(2)+4)^(11/2) * n^(9/2) * (1+sqrt(2))^(2*n-8) / (9 * 2^(19/2) * sqrt(Pi)) * (1 - 12*sqrt(2*Pi*(4+3*sqrt(2)))/(37*sqrt(n))). - Vaclav Kotesovec, Oct 27 2016

Mathematica

y[z] := z^2 - 6*z + 1; CoefficientList[Series[(2*(36*z^7 + 20*z^6 + 24*z^5 - 219*z^4 + 216*z^3 + 163*z^2 + 6*z))/(y[z]^(11/2)) + (2*(12*z^8 - 132*z^7 + 618*z^6 - 1830*z^5 + 1840*z^4 + 720*z^3 - 134*z^2 - 6*z))/(y[z]^6), {z, 0, 50}], z] (* G. C. Greubel, Jan 29 2017 *)

Crossrefs

Cf. A006318, A277661, A277662, A277663, A277665.

Sequence in context: A078399 A243478 A245658 * A199836 A196705 A196854

Adjacent sequences: A277661 A277662 A277663 * A277665 A277666 A277667

Keyword

nonn

Author

Fabio Deelan Cunden, Oct 26 2016

Extensions

More terms from Michel Marcus, Nov 01 2016

Status

approved