A166438 Number of reduced words of length n in Coxeter group on 44 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

1, 44, 1892, 81356, 3498308, 150427244, 6468371492, 278139974156, 11960018888708, 514280812214444, 22114074925221092, 950905221784506010, 40888924536733717752, 1758223755079548115128, 75603621468420493777560

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170763, 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 (42,42,42,42,42,42,42,42,42,42,-903).

Crossrefs

Cf. A154638, A169452, A170763.

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(903,42,x) )); // G. C. Greubel, Jul 26 2024

Mathematica

With[{p=903, q=42}, 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 *)
coxG[{11,903,-42}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Aug 01 2022 *)

SageMath

def f(p,q,x): return (1+x)*(1-x^11)/(1-(q+1)*x+(p+q)*x^11-p*x^12)
def A166438_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( f(903,42,x) ).list()
A166438_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)/(903*t^11 - 42*t^10 - 42*t^9 - 42*t^8 - 42*t^7 - 42*t^6 - 42*t^5 - 42*t^4 - 42*t^3 - 42*t^2 - 42*t + 1).

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

a(n) = 42*Sum_{j=1..10} a(n-j) - 903*a(n-11).

G.f.: (1+x)*(1-x^11)/(1 - 43*x + 945*x^11 - 903*x^12). (End)