A166463 Number of reduced words of length n in Coxeter group on 50 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

1, 50, 2450, 120050, 5882450, 288240050, 14123762450, 692064360050, 33911153642450, 1661646528480050, 81420679895522450, 3989613314880598825, 195491052429149282400, 9579061569028311897600, 469374016882387138922400

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170769, 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 (48,48,48,48,48,48,48,48,48,48,-1176).

Crossrefs

Cf. A154638, A169452, A170769.

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(1176,48,x) )); // G. C. Greubel, Jul 27 2024

Mathematica

With[{p=1176, q=48}, 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 15 2016; Jul 27 2024 *)
coxG[{11,1176,-48}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Apr 29 2018 *)

SageMath

def f(p,q,x): return (1+x)*(1-x^11)/(1-(q+1)*x+(p+q)*x^11-p*x^12)
def A166463_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( f(1176,48,x) ).list()
A166463_list(30) # G. C. Greubel, Jul 27 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)/(1176*t^11 - 48*t^10 - 48*t^9 - 48*t^8 - 48*t^7 - 48*t^6 - 48*t^5 - 48*t^4 - 48*t^3 - 48*t^2 - 48*t + 1).

From G. C. Greubel, Jul 27 2024: (Start)

a(n) = 48*Sum_{j=1..10} a(n-j) - 1176*a(n-11).

G.f.: (1+x)*(1-x^11)/(1 - 49*x + 1224*x^11 - 1176*x^12). (End)