A166436 Number of reduced words of length n in Coxeter group on 42 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

1, 42, 1722, 70602, 2894682, 118681962, 4865960442, 199504378122, 8179679503002, 335366859623082, 13750041244546362, 563751691026399981, 23113819332082363920, 947666592615375474240, 38854330297230335138160

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170761, 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 (40,40,40,40,40,40,40,40,40,40,-820).

Crossrefs

Cf. A154638, A169452, A170761.

Programs

Magma

R:=PowerSeriesRing(Integers(), 30);
Coefficients(R!( (1+x)*(1-x^11)/(1-41*x+860*x^11-820*x^12) )); // G. C. Greubel, Jul 26 2024

Mathematica

With[{p=820, q=40}, 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[{11,820,-40}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Apr 27 2024 *)

SageMath

def A166436_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^11)/(1-41*x+860*x^11-820*x^12) ).list()
A166436_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)/(820*t^11 - 40*t^10 - 40*t^9 - 40*t^8 - 40*t^7 - 40*t^6 - 40*t^5 - 40*t^4 - 40*t^3 - 40*t^2 - 40*t + 1).

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

a(n) = 40*Sum_{j=1..10} a(n-j) - 820*a(n-11).

G.f.: (1+x)*(1-x^11)/(1 - 41*x + 860*x^11 - 820*x^12). (End)