A166431 Number of reduced words of length n in Coxeter group on 37 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

1, 37, 1332, 47952, 1726272, 62145792, 2237248512, 80540946432, 2899474071552, 104381066575872, 3757718396731392, 135277862282329446, 4870003042163836080, 175320109517897236410, 6311523942644269461840

Offset

0,2

Comments

The initial terms coincide with those of A170756, 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 (35,35,35,35,35,35,35,35,35,35,-630).

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)/(630*t^11 - 35*t^10 - 35*t^9 - 35*t^8 - 35*t^7 - 35*t^6 - 35*t^5 - 35*t^4 - 35*t^3 - 35*t^2 - 35*t + 1).

From G. C. Greubel, Jul 25 2024: (Start)

a(n) = 35*Sum_{j=1..10} a(n-j) - 630*a(n-11).

G.f.: (1+x)*(1-x^11)/(1 - 36*x + 665*x^11 - 630*x^12). (End)

Mathematica

With[{p=630, q=35}, 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 25 2024 *)
coxG[{11, 630, -35. 30}] (* The coxG program is at A169452 *) (* G. C. Greubel, Jul 25 2024 *)

Prog

(Magma)
R<x>:=PowerSeriesRing(Integers(), 30);
Coefficients(R!( (1+x)*(1-x^11)/(1-36*x+665*x^11-630*x^12) )); // G. C. Greubel, Jul 25 2024
(SageMath)
def A166431_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^11)/(1-36*x+665*x^11-630*x^12) ).list()
A166431_list(30) # G. C. Greubel, Jul 25 2024

Crossrefs

Cf. A154638, A169452, A170756.

Sequence in context: A165169 A165654 A166167 * A166689 A167090 A167468

Adjacent sequences: A166428 A166429 A166430 * A166432 A166433 A166434

Keyword

nonn

Author

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

Status

approved