A163222 Number of reduced words of length n in Coxeter group on 39 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.

1, 39, 1482, 56316, 2139267, 81263988, 3086962281, 117263934684, 4454486050560, 169211838474861, 6427822638540342, 244172655087350379, 9275347010187982854, 352341101130365494992, 13384324210123816783899

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..630
• Index entries for linear recurrences with constant coefficients, signature (37, 37, 37, -703).

Programs

Magma

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^4)/(1-38*x+740*x^4-703*x^5) )); // G. C. Greubel, Apr 30 2019

Mathematica

CoefficientList[Series[(t^4+2*t^3+2*t^2+2*t+1)/(703*t^4-37*t^3-37*t^2 - 37*t+1), {t,0,20}], t] (* or *) Join[{1}, LinearRecurrence[{37, 37, 37, -703}, {39, 1482, 56316, 2139267}, 20]] (* G. C. Greubel, Dec 11 2016 *)
coxG[{4, 703, -37}] (* The coxG program is at A169452 *) (* G. C. Greubel, Apr 30 2019 *)

PARI

my(t='t+O('t^20)); Vec((t^4+2*t^3+2*t^2+2*t+1)/(703*t^4-37*t^3 - 37*t^2-37*t+1)) \\ G. c. Greubel, Dec 11 2016

Sage

((1+x)*(1-x^4)/(1-38*x+740*x^4-703*x^5)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 30 2019

Formula

G.f.: (t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(703*t^4 - 37*t^3 - 37*t^2 - 37*t + 1).

a(n) = 37*a(n-1)+37*a(n-2)+37*a(n-3)-703*a(n-4). - Wesley Ivan Hurt, May 06 2021