A163645 Number of reduced words of length n in Coxeter group on 37 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

1, 37, 1332, 47952, 1726272, 62145126, 2237200560, 80538357690, 2899349827920, 104375476044000, 3757476898626570, 135267719763613500, 4869585762918574950, 175303210136598476100, 6310847981816367469200

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, 35, -630).

Programs

GAP

a:=[37, 1332, 47952, 1726272, 62145126];; for n in [6..20] do a[n]:=18*(a[n-1]+a[n-2] +a[n-3]+a[n-4] -18*a[n-5]); od; Concatenation([1], a); # G. C. Greubel, May 22 2019

Magma

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^5)/(1-36*x+665*x^5-630*x^6) )); // G. C. Greubel, May 22 2019

Mathematica

CoefficientList[Series[(1+x)*(1-x^5)/(1-36*x+665*x^5-630*x^6), {x,0,20}], x] (* G. C. Greubel, Aug 01 2017 *)
coxG[{5,630,-35}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Aug 31 2018 *)

PARI

my(x='x+O('x^20)); Vec((1+x)*(1-x^5)/(1-36*x+665*x^5-630*x^6)) \\ G. C. Greubel, Aug 01 2017

Sage

((1+x)*(1-x^5)/(1-36*x+665*x^5-630*x^6)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, May 22 2019

Formula

G.f.: (t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(630*t^5 - 35*t^4 - 35*t^3 - 35*t^2 - 35*t + 1).

a(n) = -630*a(n-5) + 35*Sum_{k=1..4} a(n-k). - Wesley Ivan Hurt, May 05 2021