A167957 Number of reduced words of length n in Coxeter group on 41 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

1, 41, 1640, 65600, 2624000, 104960000, 4198400000, 167936000000, 6717440000000, 268697600000000, 10747904000000000, 429916160000000000, 17196646400000000000, 687865856000000000000, 27514634240000000000000, 1100585369600000000000000, 44023414783999999999999180

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170760, 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 (39,39,39,39,39,39,39,39,39,39,39,39,39,39,39,-780).

Crossrefs

Cf. A154638, A169452, A170760.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-40*x+819*x^16-780*x^17) )); // G. C. Greubel, Jul 14 2023

Mathematica

CoefficientList[Series[(1+t)*(1-t^16)/(1-40*t+819*t^16-780*t^17), {t, 0, 40}], t] (* G. C. Greubel, Jul 02 2016; Jul 14 2023 *)
coxG[{16,780,-39}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Nov 20 2021 *)

SageMath

def A167957_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^16)/(1-40*x+819*x^16-780*x^17) ).list()
A167957_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)/( 780*t^16 - 39*t^15 - 39*t^14 - 39*t^13 - 39*t^12 - 39*t^11 - 39*t^10 - 39*t^9 - 39*t^8 - 39*t^7 - 39*t^6 - 39*t^5 - 39*t^4 - 39*t^3 - 39*t^2 - 39*t + 1).

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

G.f.: (1+t)*(1-t^16)/(1 -40*t +819*t^16 -780*t^17).

a(n) = -780*a(n-16) + 39*Sum_{j=1..15} a(n-j). (End)