A166691 Number of reduced words of length n in Coxeter group on 39 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

1, 39, 1482, 56316, 2140008, 81320304, 3090171552, 117426518976, 4462207721088, 169563893401344, 6443427949251072, 244850262071540736, 9304309958718547227, 353563778431304766468, 13435423580389580056521

Offset: 0

Author

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

Keywords

• nonn

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, -703).

Programs

Magma

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^12)/(1-38*x+740*x^12-703*x^13) )); // G. C. Greubel, Apr 26 2019

Mathematica

CoefficientList[Series[(1+x)*(1-x^12)/(1-38*x+740*x^12-703*x^13), {x, 0, 20}], x] (* G. C. Greubel, May 23 2016, modified Apr 26 2019 *)
coxG[{12,703,-37}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jan 10 2017 *)

PARI

my(x='x+O('x^20)); Vec((1+x)*(1-x^12)/(1-38*x+740*x^12-703*x^13)) \\ G. C. Greubel, Apr 26 2019

Sage

((1+x)*(1-x^12)/(1-38*x+740*x^12-703*x^13)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 26 2019

Formula

G.f.: (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^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).

G.f.: (1+x)*(1-x^12)/(1 -38*x +740*x^12 -703*x^13). - G. C. Greubel, Apr 26 2019

a(n) = -703*a(n-12) + 37*Sum_{k=1..11} a(n-k). - Wesley Ivan Hurt, May 06 2021