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

0,2

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

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)

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 *)

Prog

(Magma) R<x>:=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
(SageMath)
def A167955_list(prec):
    P.<x> = 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

Crossrefs

Cf. A154638, A169452, A170756.

Sequence in context: A167090 A167468 A167818 * A168714 A168762 A168810

Adjacent sequences: A167950 A167951 A167952 * A167954 A167955 A167956

Keyword

nonn

Author

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

Status

approved