A007988 - OEIS

%I #34 Sep 08 2022 08:44:35

%S 2,2,5,6,11,12,20,22,32,36,49,54,71,78,98,108,132,144,173,188,221,240,

%T 278,300,344,370,419,450,505,540,602,642,710,756,831,882,965,1022,

%U 1112,1176,1274,1344,1451,1528,1643,1728,1852,1944,2078,2178

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

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

%H Vincenzo Librandi, <a href="/A007988/b007988.txt">Table of n, a(n) for n = 2..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_08">Index entries for linear recurrences with constant coefficients</a>, signature (1, 2, -1, -2, -1, 2, 1, -1).

%F 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]

%F 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

%F a(n) = floor((n+1)*(27*(-1)^n+41+16*n+2*n^2)/144). - _Tani Akinari_, Jun 26 2013

%p 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

%t 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 *)

%o (Magma) [Floor((n+1)*(27*(-1)^n+41+16*n+2*n^2)/144): n in [2..60]]; // _Vincenzo Librandi_, Mar 04 2014

%K nonn

%O 2,1

%A _Stephen P. Humphries_

%E More terms from _Ralf Stephan_, Jun 11 2005