A167953 Number of reduced words of length n in Coxeter group on 37 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

1, 37, 1332, 47952, 1726272, 62145792, 2237248512, 80540946432, 2899474071552, 104381066575872, 3757718396731392, 135277862282330112, 4870003042163884032, 175320109517899825152, 6311523942644393705472

Offset: 0

Author

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

Keywords

• nonn

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 (35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,-630).

Crossrefs

Cf. A154638, A169452, A170756.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-36*x+665*x^16-630*x^17) )); // G. C. Greubel, Sep 06 2023

Mathematica

CoefficientList[Series[(1+t)*(1-t^16)/(1-36*t+665*t^16-630*t^17), {t, 0, 50}], t] (* G. C. Greubel, Jul 02 2016; Sep 06 2023 *)
coxG[{16,630,-35}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Oct 14 2022 *)

SageMath

def A167955_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^16)/(1-36*x+665*x^16-630*x^17) ).list()
A167955_list(40) # G. C. Greubel, Sep 06 2023

Formula

G.f.: (t^16 + 2*t^15 + 2*t^14 + 2*t^13 + 2*t^12 + 2*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^16 - 35*t^15 - 35*t^14 - 35*t^13 - 35*t^12 - 35*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, Sep 06 2023: (Start)

G.f.: (1+t)*(1-t^16)/(1 - 36*t + 665*t^16 - 630*t^17).

a(n) = 35*Sum_{j=1..15} a(n-j) - 630*a(n-16). (End)