A167908 Number of reduced words of length n in Coxeter group on 10 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

1, 10, 90, 810, 7290, 65610, 590490, 5314410, 47829690, 430467210, 3874204890, 34867844010, 313810596090, 2824295364810, 25418658283290, 228767924549610, 2058911320946445, 18530201888517600, 166771816996654800

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A003952, 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..500
• Index entries for linear recurrences with constant coefficients, signature (8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,-36).

Crossrefs

Cf. A003952, A154638, A169452.

Programs

Magma

R:=PowerSeriesRing(Integers(), 30);
Coefficients(R!( (1+x)*(1-x^16)/(1-9*x+44*x^16-36*x^17) )); // G. C. Greubel, Jul 23 2024

Mathematica

With[{a=36, b=8}, CoefficientList[Series[(1+t)*(1-t^16)/(1-(b+1)*t +(a + b)*t^16 -a*t^17), {t,0,40}], t]] (* G. C. Greubel, Jul 01 2016; Jul 23 2024 *)
coxG[{16,36,-8}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jun 04 2017 *)

SageMath

def A167908_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^16)/(1-9*x+44*x^16-36*x^17) ).list()
A167908_list(30) # G. C. Greubel, Jul 23 2024

Formula

G.f.: (t^16 + 2*t^15 + 2*t^14 + 2*t^13 + 2*t^12 + 2*t^11 + 2*t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/( 36*t^16 - 8*t^15 - 8*t^14 - 8*t^13 - 8*t^12 - 8*t^11 - 8*t^10 - 8*t^9 - 8*t^8 - 8*t^7 - 8*t^6 - 8*t^5 - 8*t^4 - 8*t^3 - 8*t^2 - 8*t + 1).

From G. C. Greubel, Jul 23 2024: (Start)

a(n) = 8*Sum_{j=1..15} a(n-j) - 36*a(n-16).

G.f.: (1+t)*(1 - t^16)/(1 - 9*t + 44*t^16 - 36*t^17). (End)