A167927 Number of reduced words of length n in Coxeter group on 18 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

1, 18, 306, 5202, 88434, 1503378, 25557426, 434476242, 7386096114, 125563633938, 2134581776946, 36287890208082, 616894133537394, 10487200270135698, 178282404592306866, 3030800878069216722, 51523614927176684121

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170737, 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 (16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,-136).

Crossrefs

Cf. A154638, A169452, A170737.

Cf. A167881, A167882, A167896 - A167900, A167908, A167914, A167916, A167919, A167922, A167923, A167924, A167926, A167929, A167931, A167933, A167935, A167937, A167938, A167940 - A167947, A167949 - A167962, A167978, A167980, A167988, A167989.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-17*x+152*x^16-136*x^17) )); // G. C. Greubel, Sep 10 2023

Mathematica

CoefficientList[Series[(1+t)*(1-t^16)/(1-17*t+152*t^16-136*t^17), {t, 0, 50}], t] (* G. C. Greubel, Jul 01 2016; Sep 10 2023 *)
coxG[{16,136,-16}] (* The coxG program is at A169452 *) (* Harvey P. Dale, May 15 2022 *)

SageMath

def A167927_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^16)/(1-17*x+152*x^16-136*x^17) ).list()
A167927_list(40) # G. C. Greubel, Sep 10 2023

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)/( 136*t^16 - 16*t^15 - 16*t^14 - 16*t^13 - 16*t^12 - 16*t^11 - 16*t^10 - 16*t^9 - 16*t^8 - 16*t^7 - 16*t^6 - 16*t^5 - 16*t^4 - 16*t^3 - 16*t^2 - 16*t + 1).

From G. C. Greubel, Sep 10 2023: (Start)

G.f.: (1+t)*(1-t^16)/(1 - 17*t + 152*t^16 - 136*t^17).

a(n) = 16*Sum_{j=1..15} a(n-j) - 136*a(n-16). (End)