Fabio Deelan Cunden - cp's OEIS frontend

A277665 5th-order coefficients of the 1/N-expansion of traces of negative powers of real Wishart matrices with parameter c=2.

0, 0, 42, 6426, 291696, 7786680, 152881422, 2451889734, 34052988736, 424606263984, 4868397305884, 52193110266396, 529596113392928, 5132630490667056, 47846123752559076, 431382289963465044, 3778388016547646976, 32265703705734047808, 269434703704642529046, 2205554182120984631622

Offset: 0

Fabio Deelan Cunden, Oct 26 2016

nonn

These numbers provide the 5th 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 given by the Large Schröder numbers A006318.)

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).

Cf. A006318 (0th order), A277661 (1st order), A277662 (2nd order), A277663 (3rd order), A277664 (4th order).

y[z] := z^2 - 6*z + 1; CoefficientList[Series[-(2*z*(96*z^7 - 456*z^6 + 2992*z^5 - 7068*z^4 + 3089*z^3 + 8214*z^2 + 979*z + 12))/(y[z]^(13/2)) - (2*z*(288*z^8 + 776*z^7 - 336*z^6 - 2916*z^5 + 6276*z^4 - 1312*z^3 - 7560*z^2 - 964*z - 12))/(y[z]^7), {z,0,50}], z] (* G. C. Greubel, Jan 29 2017 *)

G.f.: -(2*z*(96*z^7 - 456*z^6 + 2992*z^5 - 7068*z^4 + 3089*z^3 + 8214*z^2 + 979*z + 12)) / (y(z)^(13/2)) - (2*z*(288*z^8 + 776*z^7 - 336*z^6 - 2916*z^5 + 6276*z^4 - 1312*z^3 - 7560*z^2 - 964*z - 12)) / (y(z)^7), where y(z) = z^2-6*z+1.

More terms from Michel Marcus, Oct 30 2016

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

Original entry on oeis.org

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

Offset: 0

Fabio Deelan Cunden, Oct 26 2016

nonn

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.)

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.

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

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 *)

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

More terms from Michel Marcus, Nov 01 2016

A277663 3rd-order coefficients of the 1/N-expansion of traces of negative powers of real Wishart matrices with parameter c=2.

Original entry on oeis.org

0, 0, 10, 378, 7048, 96000, 1092460, 11060700, 103150528, 905077728, 7576640950, 61098854454, 477942694136, 3645484792560, 27220292840440, 199588002587160, 1440630859132416, 10256896070590464, 72150109176698562, 502120765832371602, 3461203073248719400, 23654601049848668256

Offset: 0

Fabio Deelan Cunden, Oct 26 2016

nonn

These numbers provide the 3rd 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 given by the Large Schröder numbers A006318.)

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). E-print arXiv:1607.00250.
J. Kuipers, M. Sieber and D. Savin, Efficient semiclassical approach for time delays, New J. Phys. 16, 123018 (2014), arXiv:1409.1532 [nlin.CD], 2014.

Cf. A006318 (0th order), A277661 (1st order), A277662 (2nd order), A277664 (4th order), A277665 (5th order).

CoefficientList[Series[-(2 x) (2 x^3 - 9 x^2 + 19 x + 3) / ((x^2 - 6 x + 1)^(7/2)) - (2 x) (6 x^4 - 5 x^3 + 9 x^2 - 15 x - 3) / ((x^2 - 6 x + 1)^4), {x, 0, 25}], x] (* Vincenzo Librandi, Nov 07 2016 *)

G.f.: -(2*z)*(2*z^3-9*z^2+19*z+3)/(y(z)^(7/2))-(2*z)*(6*z^4-5*z^3+9*z^2-15*z-3)/(y(z)^4) where y(z)=z^2-6*z+1.

a(n) ~ (17*sqrt(2)/24-1) * n^3 * (1+sqrt(2))^(2*n+6) * (1 - (7*sqrt((8+6*sqrt(2)) / Pi))/(8*sqrt(n))). - Vaclav Kotesovec, Oct 27 2016

A277662 2nd-order coefficients of the 1/N-expansion of traces of negative powers of real Wishart matrices with parameter c=2.

Original entry on oeis.org

0, 0, 6, 102, 1142, 10650, 89576, 705012, 5297924, 38478492, 272262050, 1887071274, 12862479402, 86468603910, 574580180020, 3780504491400, 24663229376872, 159709443132888, 1027505285362590, 6572573611318158, 41827041105943870, 264959521695360786, 1671472578046156512, 10504743400858155708

Offset: 0

Fabio Deelan Cunden, Oct 26 2016

nonn

These numbers provide the 2nd 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 given by the Large Schröder numbers A006318.)

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.

Cf. A006318 (0th order), A277661 (1st order), A277663 (3rd order), A277664 (4th order), A277665 (5th order).

CoefficientList[Series[(x^2 - 3 x)/((x^2 - 6 x + 1)^2) + (3 x^3 - 4 x^2 + 3 x)/((x^2 - 6 x + 1)^(5/2)), {x, 0, 23}], x] (* Michael De Vlieger, Oct 26 2016 *)

G.f.: (x^2-3*x)/((x^2-6*x+1)^2)+(3*x^3-4*x^2+3 x)/((x^2-6*x+1)^(5/2)).

a(n) ~ 7*(3*sqrt(2)+4)^(5/2) * n^(3/2) * (1+sqrt(2))^(2*n-4) / (3*2^(9/2)*sqrt(Pi)) * (1 - (3*sqrt((2+3/sqrt(2))*Pi))/(7*sqrt(n))). - Vaclav Kotesovec, Oct 27 2016

More terms from Michael De Vlieger, Oct 26 2016

A277661 1st-order coefficients of the 1/N-expansion of traces of negative powers of real Wishart matrices with parameter c=2.

Original entry on oeis.org

0, 0, 2, 18, 128, 840, 5306, 32802, 200064, 1209168, 7261042, 43394802, 258401216, 1534310232, 9089538922, 53748310338, 317337926144, 1871206403232, 11021718519266, 64859423566290, 381371547195648, 2240888478928488, 13159108981577242, 77232197285953890, 453066998085075840, 2656691258873376240

Offset: 0

Fabio Deelan Cunden, Oct 26 2016

nonn

These numbers provide the 1st 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 given by the Large Schröder numbers A006318.)

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.

Cf. A006318 (0th order), A277662 (2nd order), A277663 (3rd order), A277664 (4th order), A277665 (5th order).

CoefficientList[Series[(1 - 3 x)/(2 (x^2 - 6 x + 1)) - 1/(2 (x^2 - 6 x + 1)^(1/2)), {x, 0, 25}], x] (* Michael De Vlieger, Oct 26 2016 *)

G.f.: (1-3*x)/(2*(x^2-6*x+1))-1/(2*(x^2-6*x+1)^(1/2)).

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

More terms from Michael De Vlieger, Oct 26 2016

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

Original entry on oeis.org

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

Fabio Deelan Cunden, Oct 26 2016

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.)

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.

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

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

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 *)

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

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)

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

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

A272865 Triangle read by rows, T(n,k) are covariances of inverse power traces of complex Wishart matrices with parameter c=2, for n>=1 and 1<=k<=n.

Original entry on oeis.org

4, 24, 160, 132, 936, 5700, 720, 5312, 33264, 198144, 3940, 29880, 190980, 1155600, 6823620, 21672, 167712, 1088856, 6670656, 39786120, 233908896, 119812, 941640, 6189540, 38300976, 230340740, 1363667256, 7997325700

Offset: 1

Fabio Deelan Cunden, May 08 2016

These numbers provide the covariances of power traces of the time-delay matrix when the scattering matrix belongs to the Dyson ensembles.

Relation with A047781 and A002002. See eq. (60) and (61) in Cunden et al., J. Phys. A: Math. Theor. 49, 18LT01 (2016).

			Triangle starts:
4;
24,   160;
132,  936,   5700;
720,  5312,  33264,  198144;
3940, 29880, 190980, 1155600, 6823620;

F. D. Cunden, "Statistical distribution of the Wigner-Smith time-delay matrix moments for chaotic cavities", Phys. Rev. E 91, 060102(R) (2015).
F. D. Cunden, F. Mezzadri, N. Simm and P. Vivo, "Correlators for the Wigner-Smith time-delay matrix of chaotic cavities", J. Phys. A: Math. Theor. 49, 18LT01 (2016).
F. D. Cunden, F. Mezzadri, N. O'Connell and N. Simm, "Moments of Random Matrices and Hypergeometric Orthogonal Polynomials", Commun. Math. Phys. 369, 1091-1145 (2019).

F. D. Cunden, Statistical distribution of the Wigner-Smith time-delay matrix moments for chaotic cavities, arXiv:1412.2172 [cond-mat.mes-hall], 2014-2015.
F. D. Cunden, F. Mezzadri, N. Simm and P. Vivo, Correlators for the Wigner-Smith time-delay matrix of chaotic cavities, arXiv:1601.06690 [math-ph], 2016.
F. D. Cunden, F. Mezzadri, N. O'Connell and N. Simm, Moments of Random Matrices and Hypergeometric Orthogonal Polynomials, arXiv:1805.08760 [math-ph], 2018.

Cf. A002002, A047781, A050151.

P := (n,k) -> simplify(n*hypergeom([1-k,k+1],[1],-1)*hypergeom([1-n,n+1],[2],-1)): seq(seq(4*(n*k)*(P(n,k)+P(k,n))/(n+k),k=1..n),n=1..7); # Peter Luschny, May 08 2016

Clear["Global`*"];(*Wigner-Smith Covariance*)
P[k_] := Sum[Binomial[k - 1, j] Binomial[k + j, j], {j, 0, k - 1}]
Q[k_] := Sum[Binomial[k, j + 1] Binomial[k + j, j], {j, 0, k - 1}]
a[k1_, k2_] := 4 (k1 k2)/(k1 + k2) (P[k1] Q[k2] + P[k2] Q[k1])
L = 10; Table[a[k, l], {k, 1, L}, {l, 1, k}]

G.f.: ((x*y)/(x-y)^2)*((x*y-3(x+y)+1)/(sqrt(x^2-6x+1)*sqrt(y^2-6y+1))-1).

T(n,1)/4 = A050151(n) for n>=1. - Peter Luschny, May 08 2016

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User: Fabio Deelan Cunden

A277665 5th-order coefficients of the 1/N-expansion of traces of negative powers of real Wishart matrices with parameter c=2.

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

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A277663 3rd-order coefficients of the 1/N-expansion of traces of negative powers of real Wishart matrices with parameter c=2.

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A277662 2nd-order coefficients of the 1/N-expansion of traces of negative powers of real Wishart matrices with parameter c=2.

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A277661 1st-order coefficients of the 1/N-expansion of traces of negative powers of real Wishart matrices with parameter c=2.

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

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A272865 Triangle read by rows, T(n,k) are covariances of inverse power traces of complex Wishart matrices with parameter c=2, for n>=1 and 1<=k<=n.

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