A166420 Number of reduced words of length n in Coxeter group on 26 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

1, 26, 650, 16250, 406250, 10156250, 253906250, 6347656250, 158691406250, 3967285156250, 99182128906250, 2479553222655925, 61988830566390000, 1549720764159547200, 38743019103983610000, 968575477599463500000

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170745, 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 (24,24,24,24,24,24,24,24,24,24,-300).

Crossrefs

Cf. A154638, A169452, A170745.

Programs

Magma

R:=PowerSeriesRing(Integers(), 30);
Coefficients(R!( (1+x)*(1-x^11)/(1-25*x+324*x^11-300*x^12) )); // G. C. Greubel, Jul 25 2024

Mathematica

With[{p=300, q=24}, 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 13 2016; Jul 25 2024 *)
coxG[{11,300,-24}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Dec 31 2017 *)

SageMath

def A166420_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^11)/(1-25*x+324*x^11-300*x^12) ).list()
A166420_list(30) # G. C. Greubel, Jul 25 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)/(300*t^11 - 24*t^10 - 24*t^9 - 24*t^8 - 24*t^7 - 24*t^6 - 24*t^5 - 24*t^4 - 24*t^3 - 24*t^2 - 24*t + 1).

From G. C. Greubel, Jan 17 2023: (Start)

a(n) = 24*Sum_{j=1..10} a(n-j) - 300*a(n-11).

G.f.: (1+x)*(1-x^11)/(1 - 25*x + 324*x^11 - 300*x^12). (End)