A163208 Number of reduced words of length n in Coxeter group on 30 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.

1, 30, 870, 25230, 731235, 21193200, 614237400, 17802288000, 515959239390, 14953916974920, 433405617680280, 12561286100120520, 364060598322527820, 10551476830837383840, 305810801346502707360, 8863237603561904401440

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

The initial terms coincide with those of A170749, 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..680
• Index entries for linear recurrences with constant coefficients, signature (28, 28, 28, -406).

Programs

GAP

a:=[30,870,25230,731235];; for n in [5..20] do a[n]:=28*(a[n-1] + a[n-2]+a[n-3]) -406*a[n-4]; od; Concatenation([1], a); # G. C. Greubel, Apr 28 2019

Magma

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^4)/(1-29*x+434*x^4-406*x^5) )); // G. C. Greubel, Apr 28 2019

Mathematica

CoefficientList[Series[(t^4+2*t^3+2*t^2+2*t+1)/(406*t^4-28*t^3-28*t^2- 28*t+1), {t,0,20}], t] (* or *) LinearRecurrence[{28,28,28,-406}, {1,30, 870,25230,731235}, 20] (* G. C. Greubel, Dec 10 2016 *)
coxG[{4, 406, -28}] (* The coxG program is at A169452 *) (* G. C. Greubel, Apr 28 2019 *)

PARI

my(x='x+O('x^20)); Vec((1+x)*(1-x^4)/(1-29*x+434*x^4-406*x^5)) \\ G. C. Greubel, Dec 10 2016, modified Apr 28 2019

Sage

((1+x)*(1-x^4)/(1-29*x+434*x^4-406*x^5)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 28 2019

Formula

G.f.: (t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(406*t^4 - 28*t^3 - 28*t^2 - 28*t + 1).

From G. C. Greubel, Apr 28 2019: (Start)

a(n) = 28*(a(n-1) + a(n-2) + a(n-3)) - 406*a(n-4).

G.f.: (1+x)*(1-x^4)/(1 - 29*x + 434*x^4 - 406*x^5). (End)