A163527 Number of reduced words of length n in Coxeter group on 27 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

1, 27, 702, 18252, 474552, 12338001, 320778900, 8340014475, 216834216300, 5637529462500, 146571601954050, 3810753388040625, 99076773337132500, 2575922925294444375, 66972093393463976250, 1741224960366454777500

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170746, 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..700
• Index entries for linear recurrences with constant coefficients, signature (25, 25, 25, 25, -325).

Programs

Magma

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^5)/(1-26*x+350*x^5-325*x^6) )); // G. C. Greubel, May 16 2019

Mathematica

CoefficientList[Series[(1+x)*(1-x^5)/(1-26*x+350*x^5-325*x^6), {x, 0, 20}], x] (* G. C. Greubel, Jul 27 2017 *)
coxG[{5, 325, -25}] (* The coxG program is at A169452 *) (* G. C. Greubel, May 16 2019 *)

PARI

my(x='x+O('x^20)); Vec((1+x)*(1-x^5)/(1-26*x+350*x^5-325*x^6)) \\ G. C. Greubel, Jul 27 2017

Sage

((1+x)*(1-x^5)/(1-26*x+350*x^5-325*x^6)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, May 16 2019

Formula

G.f.: (t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(325*t^5 - 25*t^4 - 25*t^3 - 25*t^2 - 25*t + 1).

a(n) = 25*a(n-1)+25*a(n-2)+25*a(n-3)+25*a(n-4)-325*a(n-5). - Wesley Ivan Hurt, May 10 2021