A166435 Number of reduced words of length n in Coxeter group on 41 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

1, 41, 1640, 65600, 2624000, 104960000, 4198400000, 167936000000, 6717440000000, 268697600000000, 10747904000000000, 429916159999999180, 17196646399999934400, 687865855999996064820, 27514634239999790145600

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170760, 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 (39,39,39,39,39,39,39,39,39,39,-780).

Crossrefs

Cf. A154638, A169452, A170760.

Programs

Magma

R:=PowerSeriesRing(Integers(), 30);
Coefficients(R!( (1+x)*(1-x^11)/(1-40*x+819*x^11-780*x^12) )); // G. C. Greubel, Jul 26 2024

Mathematica

With[{p=780, q=39}, 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, 780, -39, 30}] (* The coxG program is at A169452 *) (* G. C. Greubel, Jul 26 2024 *)

SageMath

def A166435_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^11)/(1-40*x+819*x^11-780*x^12) ).list()
A166435_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)/(780*t^11 - 39*t^10 - 39*t^9 - 39*t^8 - 39*t^7 - 39*t^6 - 39*t^5 - 39*t^4 - 39*t^3 - 39*t^2 - 39*t + 1).

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

a(n) = 39*Sum_{j=1..10} a(n-j) - 780*a(n-11).

G.f.: (1+x)*(1-x^11)/(1 - 40*x + 819*x^11 - 780*x^12). (End)