A168815 Number of reduced words of length n in Coxeter group on 42 generators S_i with relations (S_i)^2 = (S_i S_j)^19 = I.

1, 42, 1722, 70602, 2894682, 118681962, 4865960442, 199504378122, 8179679503002, 335366859623082, 13750041244546362, 563751691026400842, 23113819332082434522, 947666592615379815402, 38854330297230572431482

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

The initial terms coincide with those of A170761, although the two sequences are eventually different.

First disagreement at index 19: a(19) = 4501515100636334942966837319021, A170761(19) = 4501515100636334942966837319882. - Klaus Brockhaus, Apr 01 2011

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 (40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, -820).

Crossrefs

Cf. A170761 (G.f.: (1+x)/(1-41*x)).

Programs

Mathematica

CoefficientList[Series[(t^19 +2*t^18 +2*t^17 +2*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)/(820*t^19 -40*t^18 -40*t^17 -40*t^16 -40*t^15 -40*t^14 - 40*t^13 -40*t^12 -40*t^11 -40*t^10 -40*t^9 -40*t^8 -40*t^7 -40*t^6 -40*t^5 - 40*t^4 -40*t^3 -40*t^2 -40*t +1), {t,0,50}], t] (* G. C. Greubel, Nov 21 2016 *)
coxG[{19,820,-40}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Sep 07 2023 *)

Formula

G.f.: (t^19 + 2*t^18 + 2*t^17 + 2*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)/(820*t^19 - 40*t^18 - 40*t^17 - 40*t^16 - 40*t^15 - 40*t^14 - 40*t^13 - 40*t^12 - 40*t^11 - 40*t^10 - 40*t^9 - 40*t^8 - 40*t^7 - 40*t^6 - 40*t^5 - 40*t^4 - 40*t^3 - 40*t^2 - 40*t + 1).