A166442 Number of reduced words of length n in Coxeter group on 48 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

1, 48, 2256, 106032, 4983504, 234224688, 11008560336, 517402335792, 24317909782224, 1142941759764528, 53718262708932816, 2524758347319841224, 118663642324032484512, 5577191189229524281440, 262127985893787524168352

Offset: 0

Author

John Cannon and N. J. A. Sloane, Dec 03 2009

Keywords

• nonn

Comments

The initial terms coincide with those of A170767, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..500
• Index entries for linear recurrences with constant coefficients, signature (46,46,46,46,46,46,46,46,46,46,-1081).

Crossrefs

Cf. A154638, A169452, A170767.

Programs

Magma

R:=PowerSeriesRing(Integers(), 30);
f:= func< p,q,x | (1+x)*(1-x^11)/(1-(q+1)*x+(p+q)*x^11-p*x^12) >;
Coefficients(R!( f(1081,46,x) )); // G. C. Greubel, Jul 26 2024

Mathematica

With[{num=Total[2t^Range[10] ]+t^11+1,den=Total[-46 t^Range[10]]+ 1081t^11+ 1}, CoefficientList[Series[num/den,{t,0,30}],t]] (* Harvey P. Dale, Jul 21 2011 *)
CoefficientList[Series[(x^11 + 2 x^10 + 2 x^9 + 2 x^8 + 2 x^7 + 2 x^6 + 2 x^5 + 2 x^4 + 2 x^3 + 2 x^2 + 2 x + 1)/(1081 x^11 - 46 x^10 - 46 x^9 - 46 x^8 - 46 x^7 - 46 x^6 - 46 x^5 - 46 x^4 - 46 x^3 - 46 x^2 - 46 x + 1), {x, 0, 20}], x] (* Vincenzo Librandi, May 10 2015 *)
coxG[{11, 1081, -46, 30}] (* The coxG program is at A169452 *) (* G. C. Greubel, Jul 26 2024 *)

SageMath

def f(p,q,x): return (1+x)*(1-x^11)/(1-(q+1)*x+(p+q)*x^11-p*x^12)
def A166442_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( f(1081,46,x) ).list()
A166442_list(30) # G. C. Greubel, Jul 26 2024

Formula

G.f.: (t^11 + 2*t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(1081*t^11 - 46*t^10 - 46*t^9 - 46*t^8 - 46*t^7 - 46*t^6 - 46*t^5 - 46*t^4 - 46*t^3 - 46*t^2 - 46*t + 1).

From G. C. Greubel, Jul 26 2024: (Start)

a(n) = 46*Sum_{j=1..10} a(n-j) - 1081*a(n-11).

G.f.: (1+x)*(1-x^11)/(1 - 47*x + 1127*x^11 - 1081*x^12). (End)