A007988 Expansion of (x^6-x^5-x^4+2x^2)/((1-x^3)(1-x^2)^2(1-x)).

2, 2, 5, 6, 11, 12, 20, 22, 32, 36, 49, 54, 71, 78, 98, 108, 132, 144, 173, 188, 221, 240, 278, 300, 344, 370, 419, 450, 505, 540, 602, 642, 710, 756, 831, 882, 965, 1022, 1112, 1176, 1274, 1344, 1451, 1528, 1643, 1728, 1852, 1944, 2078, 2178

Offset

2,1

Comments

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 2..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 (1, 2, -1, -2, -1, 2, 1, -1).

Formula

a(n) = -25/72+A000217(n+1)/12+A000292(n+1)/12+17*(n+1)/144+3*(n+1)*(-1)^n/16-2*A049347(n+2)/9-(-1)^n/8. - R. J. Mathar, Apr 23 2009

a(2)=2, a(3)=2, a(4)=5, a(5)=6, a(6)=11, a(7)=12, a(8)=20, a(9)=22; for n>9, a(n) = a(n-1)+ 2*a(n-2)-a(n-3)-2*a(n-4)-a(n-5)+2*a(n-6)+a(n-7)-a(n-8). - Harvey P. Dale, Apr 04 2013

a(n) = floor((n+1)*(27*(-1)^n+41+16*n+2*n^2)/144). - Tani Akinari, Jun 26 2013

a(n) ~ n^3/72. - Charles R Greathouse IV, May 31 2026

Maple

A007988:=n->floor((n+1)*(27*(-1)^n+41+16*n+2*n^2)/144); seq(A007988(n), n=2..100); # Wesley Ivan Hurt, Feb 26 2014

Mathematica

Drop[CoefficientList[Series[(x^6-x^5-x^4+2x^2)/((1-x^3)(1-x^2)^2(1-x)), {x, 0, 60}], x], 2] (* or *) LinearRecurrence[{1, 2, -1, -2, -1, 2, 1, -1}, {2, 2, 5, 6, 11, 12, 20, 22}, 60] (* Harvey P. Dale, Apr 04 2013 *)

Prog

(Magma) [Floor((n+1)*(27*(-1)^n+41+16*n+2*n^2)/144): n in [2..60]]; // Vincenzo Librandi, Mar 04 2014
(PARI) a(n)=(n+1)*(27*(-1)^n+41+16*n+2*n^2)\144 \\ Charles R Greathouse IV, May 31 2026

Crossrefs

Cf. A000217, A000292, A049347.

Sequence in context: A288766 A348324 A377368 * A241449 A240184 A317853

Adjacent sequences: A007985 A007986 A007987 * A007989 A007990 A007991

Keyword

nonn,easy

Author

Stephen P. Humphries

Extensions

More terms from Ralf Stephan, Jun 11 2005

Status

approved