A167958 Number of reduced words of length n in Coxeter group on 42 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = 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

Comments

The initial terms coincide with those of A170761, 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 (40,40,40,40,40,40,40,40,40,40,40,40,40,40,40,-820).

Crossrefs

Cf. A154638, A169452, A170761.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-41*x+860*x^16-820*x^17) )); // G. C. Greubel, Jul 14 2023

Mathematica

CoefficientList[Series[(1+t)*(1-t^16)/(1-41*t+860*t^16 -820*t^17), {t, 0, 40}], t] (* G. C. Greubel, Jul 02 2016; Jul 14 2023 *)
coxG[{16, 820, -40, 30}] (* The coxG program is at A169452 *) (* G. C. Greubel, Jul 14 2023 *)

SageMath

def A167958_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^16)/(1-41*x+860*x^16-820*x^17) ).list()
A167958_list(40) # G. C. Greubel, Jul 14 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)/( 820*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).

From G. C. Greubel, Jul 14 2023: (Start)

G.f.: (1 + t)*(1 - t^16)/(1 - 41*t + 860*t^16 - 820*t^17).

a(n) = -820*a(n-16) + 40*Sum_{j=1..15} a(n-j). (End)