A166443 Number of reduced words of length n in Coxeter group on 49 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

1, 49, 2352, 112896, 5419008, 260112384, 12485394432, 599298932736, 28766348771328, 1380784741023744, 66277667569139712, 3181328043318705000, 152703746079297783552, 7329779811806290902168, 351829430966701833304320

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170768, 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 (47,47,47,47,47,47,47,47,47,47,-1128).

Crossrefs

Cf. A154638, A169452, A170768.

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(1128,47,x) )); // G. C. Greubel, Jul 27 2024

Mathematica

With[{num=Total[2t^Range[10]]+t^11+1,den=Total[-47 t^Range[10]]+ 1128t^11+ 1}, CoefficientList[Series[num/den,{t,0,20}],t]] (* Harvey P. Dale, Aug 29 2011 *)
With[{p=1128, q=47}, 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 27 2024 *)
coxG[{11, 1128, -47, 30}] (* The coxG program is at A169452 *) (* G. C. Greubel, Jul 27 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 A166443_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( f(1128,47,x) ).list()
A166443_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)/(1128*t^11 - 47*t^10 - 47*t^9 - 47*t^8 - 47*t^7 - 47*t^6 - 47*t^5 - 47*t^4 - 47*t^3 - 47*t^2 - 47*t + 1).

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

a(n) = 47*Sum_{j=1..10} a(n-j) - 1128*a(n-11).

G.f.: (1+x)*(1-x^11)/(1 - 48*x + 1175*x^11 - 1128*x^12). (End)