A167125 Number of reduced words of length n in Coxeter group on 18 generators S_i with relations (S_i)^2 = (S_i S_j)^14 = I.

1, 18, 306, 5202, 88434, 1503378, 25557426, 434476242, 7386096114, 125563633938, 2134581776946, 36287890208082, 616894133537394, 10487200270135698, 178282404592306713, 3030800878069211520, 51523614927176551776

Offset

0,2

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, -136).

Formula

G.f.: (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^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).

Mathematica

coxG[{14, 136, -16}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jul 17 2015 *)
CoefficientList[Series[(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^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), {t, 0, 50}], t] (* G. C. Greubel, Jun 04 2016 *)

Prog

(PARI) first(n)=Vec((1+x^2+x^4+x^6+x^8+x^10+x^12)*(1+x)^2/(1-16*x-16*x^2-16*x^3-16*x^4-16*x^5-16*x^6-16*x^7-16*x^8-16*x^9-16*x^10-16*x^11-16*x^12-16*x^13+136*x^14)+O(x^(n+1))) \\ Charles R Greathouse IV, Jun 13 2026

Crossrefs

Cf. A170737, A154638.

Sequence in context: A166412 A166599 A167048 * A167674 A167927 A168695

Adjacent sequences: A167122 A167123 A167124 * A167126 A167127 A167128

Keyword

nonn,easy

Author

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

Status

approved