A245455 Number of minimax elements in the affine Weyl group of the Lie algebra so(2n).

1, 3, 4, 9, 23, 61, 166, 459, 1284, 3623, 10292, 29395, 84327, 242807, 701314, 2031085, 5895951, 17150013, 49975428, 145862571, 426337773, 1247741271, 3655973226, 10723668081, 31485145902, 92524150845, 272120203908, 800931753629, 2359038637409, 6952768502473

Offset: 1

Author

Peter Bala, Jul 22 2014

Keywords

• nonn
• easy

Comments

See A005773 for the number of minimax elements in the affine Weyl group of the Lie algebra so(2n+1).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• D. I. Panyushev, Ideals of Heisenberg type and minimax elements of affine Weyl groups, arXiv:math/0311347 [math.RT], Lie Groups and Invariant Theory, Amer. Math. Soc. Translations, Series 2, Volume 213, (2005), ed. E. Vinberg

Crossrefs

Cf. A005773.

Programs

Maple

A245455 := proc(n)
    coeftayl(x/2*(1+2*x)*(1+sqrt(1-2*x-3*x^2)/(1-3*x)), x=0, n);
end proc:
seq(A245455(n), n=1..30); # Wesley Ivan Hurt, Jul 26 2014

Mathematica

Rest[CoefficientList[Series[x/2*(1+2*x)*(1+Sqrt[1-2*x-3*x^2]/(1-3*x)), {x, 0, 20}], x]] (* Vaclav Kotesovec, Jul 25 2014 *)

Formula

a(n) = A005773(n-1) + 2*A005773(n-2).

O.g.f.: x/2*(1+2*x)*( 1 + sqrt(1-2*x-3*x^2)/(1-3*x) ).

a(n) ~ 5*3^(n-5/2) / sqrt(Pi*n). - Vaclav Kotesovec, Jul 25 2014

(-n+1)*a(n) +4*(1)*a(n-1) +7*(n-3)*a(n-2) +6*(n-5)*a(n-3)=0. - R. J. Mathar, Sep 06 2016

(5*n-4)*(n-1)*a(n) +2*(-5*n^2+9*n-10)*a(n-1) -3*(5*n+1)*(n-4)*a(n-2)=0. - R. J. Mathar, Sep 06 2016