A007993 Poincaré series [or Poincare series] of Lie algebra associated with a certain braid group.

3, 12, 40, 102, 219, 419, 738, 1221, 1923, 2910, 4260, 6064, 8427, 11469, 15326, 20151, 26115, 33408, 42240, 52842, 65467, 80391, 97914, 118361, 142083, 169458, 200892, 236820, 277707, 324049, 376374, 435243, 501251, 575028, 657240

Offset

1,1

Comments

The series in the Humphries paper has zeros interleaved.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

S. P. Humphries, Home page

S. P. Humphries, Braid groups, infinite Lie algebras of Cartan type and rings of invariants, Topology and its Applications, 95 (3) (1999) pp. 173-205.

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Formula

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

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

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

Mathematica

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

Crossrefs

Sequence in context: A237036 A034956 A032093 * A293366 A327319 A080929

Adjacent sequences: A007990 A007991 A007992 * A007994 A007995 A007996

Keyword

nonn,easy

Author

Stephen P. Humphries

Extensions

More terms from Ralf Stephan, Jun 11 2005

Status

approved