A166441 Number of reduced words of length n in Coxeter group on 47 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

1, 47, 2162, 99452, 4574792, 210440432, 9680259872, 445291954112, 20483429889152, 942237774900992, 43342937645445632, 1993775131690497991, 91713656057762857860, 4218828178657089175245, 194066096218225996890780

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170766, 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 (45,45,45,45,45,45,45,45,45,45,-1035).

Crossrefs

Cf. A154638, A169452, A170766.

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(1035,45,x) )); // G. C. Greubel, Jul 26 2024

Mathematica

coxG[{11,1035,-45}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jan 16 2015 *)
With[{p=1035, q=45}, CoefficientList[Series[(1+t)*(1-t^11)/(1 - (q+1)*t + (p+q)*t^11 - p*t^12), {t,0,40}], t]] (* G. C. Greubel, May 14 2016; 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 A166441_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( f(1035,45,x) ).list()
A166441_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)/(1035*t^11 - 45*t^10 - 45*t^9 - 45*t^8 - 45*t^7 - 45*t^6 - 45*t^5 - 45*t^4 - 45*t^3 - 45*t^2 - 45*t + 1).

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

a(n) = 45*Sum_{j=1..10} a(n-j) - 1035*a(n-11).

G.f.: (1+x)*(1-x^11)/(1 - 46*x + 1080*x^11 - 1035*x^12). (End)