%I #30 Jul 24 2022 17:33:51

%S 3,12,40,102,219,419,738,1221,1923,2910,4260,6064,8427,11469,15326,

%T 20151,26115,33408,42240,52842,65467,80391,97914,118361,142083,169458,

%U 200892,236820,277707,324049,376374,435243,501251,575028,657240

%N Poincaré series [or Poincare series] of Lie algebra associated with a certain braid group.

%C The series in the Humphries paper has zeros interleaved.

%H Vincenzo Librandi, <a href="/A007993/b007993.txt">Table of n, a(n) for n = 1..1000</a>

%H S. P. Humphries, <a href="http://www.math.byu.edu/~steve/">Home page</a>

%H S. P. Humphries, <a href="http://dx.doi.org/10.1016/S0166-8641(98)00007-8">Braid groups, infinite Lie algebras of Cartan type and rings of invariants</a>, Topology and its Applications, 95 (3) (1999) pp. 173-205.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).

%F a(n) = 3 + 9(n-1) + 19(n-1)(n-2)/2 + 15(n-1)(n-2)(n-3)/6 + 6(n-1)(n-2)(n-3)(n-4)/24 + (n-1)(n-2)(n-3)(n-4)(n-5)/120. - _John W. Layman_, May 12 1999

%F a(n-1) = (1/120)(n^5 + 10n^4 + 35n^3 - 10n^2 - 396n + 720) with n>1. - _Ralf Stephan_, Jun 11 2005

%F G.f. -x*(-3+6*x-13*x^2+18*x^3-12*x^4+3*x^5) / (x-1)^6 . - R. J. Mathar, Dec 02 2011

%t CoefficientList[ Series[(3 - 6x + 13x^2 - 18x^3 + 12x^4 - 3x^5) / (1 - 6x + 15x^2 - 20x^3 + 15x^4 - 6x^5 + x^6), {x, 0, 34}], x] (* _Jean-François Alcover_, Dec 02 2011 *)

%t LinearRecurrence[{6,-15,20,-15,6,-1},{3,12,40,102,219,419},40] (* _Harvey P. Dale_, Jul 24 2022 *)

%K nonn,easy

%O 1,1

%A _Stephen P. Humphries_

%E More terms from _Ralf Stephan_, Jun 11 2005