A163220 Number of reduced words of length n in Coxeter group on 37 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.

1, 37, 1332, 47952, 1725606, 62097840, 2234659770, 80416702800, 2893883982570, 104139615440700, 3747579228757350, 134860782963557700, 4853114416362432150, 174644689291688511000, 6284782282271390399250

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..640
• Index entries for linear recurrences with constant coefficients, signature (35, 35, 35, -630).

Programs

Magma

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^4)/(1-36*x+665*x^4-630*x^5) )); // G. C. Greubel, Apr 30 2019

Mathematica

CoefficientList[Series[(t^4+2*t^3+2*t^2+2*t+1)/(630*t^4-35*t^3-35*t^2 - 35*t+1), {t,0,20}], t] (* or *) LinearRecurrence[{35, 35, 35, -630}, {1, 37, 1332, 47952}, 20] (* G. C. Greubel, Dec 11 2016 *)
coxG[{4, 630, -35}] (* 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)/(630*t^4-35*t^3 - 35*t^2-35*t+1)) \\ G. C. Greubel, Dec 11 2016

Sage

((1+x)*(1-x^4)/(1-36*x+665*x^4-630*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)/(630*t^4 - 35*t^3 - 35*t^2 - 35*t + 1).

a(n) = -630*a(n-4) + 35*Sum_{k=1..3} a(n-k). - Wesley Ivan Hurt, May 05 2021