A167117 Number of reduced words of length n in Coxeter group on 16 generators S_i with relations (S_i)^2 = (S_i S_j)^14 = I.

1, 16, 240, 3600, 54000, 810000, 12150000, 182250000, 2733750000, 41006250000, 615093750000, 9226406250000, 138396093750000, 2075941406250000, 31139121093749880, 467086816406246400, 7006302246093669120

Offset

0,2

Comments

The initial terms coincide with those of A170735, 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 (14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, -105).

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)/(105*t^14 - 14*t^13 - 14*t^12 - 14*t^11 - 14*t^10 - 14*t^9 - 14*t^8 - 14*t^7 - 14*t^6 - 14*t^5 - 14*t^4 - 14*t^3 - 14*t^2 - 14*t + 1).

Mathematica

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)/ (105*t^14 - 14*t^13 - 14*t^12 - 14*t^11 - 14*t^10 - 14*t^9 - 14*t^8 - 14*t^7 - 14*t^6 - 14*t^5 - 14*t^4 - 14*t^3 - 14*t^2 - 14*t + 1), {t, 0, 50}], t] (* G. C. Greubel, Jun 04 2016 *)
coxG[{14, 105, -14}] (* The coxG program is at A169452 *) (* Harvey P. Dale, May 23 2017 *)

Prog

(PARI) first(n)=Vec((1+x^2+x^4+x^6+x^8+x^10+x^12)*(1+x)^2/(1-14*x-14*x^2-14*x^3-14*x^4-14*x^5-14*x^6-14*x^7-14*x^8-14*x^9-14*x^10-14*x^11-14*x^12-14*x^13+105*x^14)+O(x^(n+1))) \\ Charles R Greathouse IV, Jun 13 2026

Crossrefs

Cf. A170735, A154638.

Sequence in context: A166410 A166584 A167026 * A167672 A167924 A168693

Adjacent sequences: A167114 A167115 A167116 * A167118 A167119 A167120

Keyword

nonn,easy

Author

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

Status

approved