A163439 Number of reduced words of length n in Coxeter group on 14 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

1, 14, 182, 2366, 30758, 399763, 5195736, 67529280, 877681896, 11407280976, 148261073142, 1926957516120, 25044775341768, 325508355356184, 4230650423530440, 54986001777229068, 714656161291232160

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

The initial terms coincide with those of A170733, 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..895
• Index entries for linear recurrences with constant coefficients, signature (12,12,12,12,-78).

Programs

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^5)/(1-13*x+90*x^5-78*x^6) )); // G. C. Greubel, May 12 2019

Mathematica

CoefficientList[Series[(1+x)*(1-x^5)/(1-13*x+90*x^5-78*x^6), {x, 0, 30}], x] (* or *) LinearRecurrence[{12, 12, 12, 12, -78}, {1, 14, 182, 2366, 30758, 399763}, 30] (* G. C. Greubel, Dec 23 2016 *)
coxG[{5, 78, -12}] (* The coxG program is at A169452 *) (* G. C. Greubel, May 12 2019 *)

PARI

my(x='x+O('x^30)); Vec((1+x)*(1-x^5)/(1-13*x+90*x^5-78*x^6)) \\ G. C. Greubel, Dec 23 2016

Sage

((1+x)*(1-x^5)/(1-13*x+90*x^5-78*x^6)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 12 2019

Formula

G.f.: (t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(78*t^5 - 12*t^4 - 12*t^3 - 12*t^2 - 12*t + 1).

a(n) = 12*a(n-1)+12*a(n-2)+12*a(n-3)+12*a(n-4)-78*a(n-5). - Wesley Ivan Hurt, May 10 2021