A167955 Number of reduced words of length n in Coxeter group on 39 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

1, 39, 1482, 56316, 2140008, 81320304, 3090171552, 117426518976, 4462207721088, 169563893401344, 6443427949251072, 244850262071540736, 9304309958718547968, 353563778431304822784, 13435423580389583265792

Offset

0,2

Comments

The initial terms coincide with those of A170758, 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 (37,37,37,37,37,37,37,37,37,37,37,37,37,37,37,-703).

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)/( 703*t^16 - 37*t^15 - 37*t^14 - 37*t^13 - 37*t^12 - 37*t^11 - 37*t^10 - 37*t^9 - 37*t^8 - 37*t^7 - 37*t^6 - 37*t^5 - 37*t^4 - 37*t^3 - 37*t^2 - 37*t + 1).

From G. C. Greubel, Sep 06 2023: (Start)

G.f.: (1+t)*(1-t^16)/(1 - 38*t + 740*t^16 - 703*t^17).

a(n) = 37*Sum_{j=1..15} a(n-j) - 703*a(n-16). (End)

Mathematica

CoefficientList[Series[(1+t)*(1-t^16)/(1-38*t+740*t^16-703*t^17), {t, 0, 50}], t] (* G. C. Greubel, Jul 02 2016; Sep 06 2023 *)
coxG[{16, 703, -37, 40}] (* The coxG program is at A169452 *) (* G. C. Greubel, Sep 06 2023 *)

Prog

(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-38*x+740*x^16-703*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)/(1-38*x+740*x^16-703*x^17) ).list()
A167955_list(40) # G. C. Greubel, Sep 06 2023

Crossrefs

Cf. A154638, A169452, A170758.

Sequence in context: A167092 A167537 A167828 * A168716 A168764 A168812

Adjacent sequences: A167952 A167953 A167954 * A167956 A167957 A167958

Keyword

nonn

Author

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

Status

approved