A167959 Number of reduced words of length n in Coxeter group on 43 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

1, 43, 1806, 75852, 3185784, 133802928, 5619722976, 236028364992, 9913191329664, 416354035845888, 17486869505527296, 734448519232146432, 30846837807750150144, 1295567187925506306048, 54413821892871264854016

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170762, 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 (41,41,41,41,41,41,41,41,41,41, 41,41,41,41,41,-861).

Crossrefs

Cf. A154638, A169452, A170762.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1+x^16)/(1-42*x+861*x^16-820*x^17) )); // G. C. Greubel, Apr 27 2023

Mathematica

CoefficientList[Series[(1+x)*(1+x^16)/(1-42*x+861*x^16-820*x^17), {x, 0, 50}], x] (* G. C. Greubel, Jul 02 2016; Apr 27 2023 *)
coxG[{16, 861, -41, 40}] (* The coxG program is at A169452 *) (* G. C. Greubel, Apr 27 2023 *)

SageMath

def A167959_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1+x^16)/(1-42*x+861*x^16-820*x^17) ).list()
A167959_list(40) # G. C. Greubel, Apr 27 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)/( 861*t^16 - 41*t^15 - 41*t^14 - 41*t^13 - 41*t^12 - 41*t^11 - 41*t^10 - 41*t^9 - 41*t^8 - 41*t^7 - 41*t^6 - 41*t^5 - 41*t^4 - 41*t^3 - 41*t^2 - 41*t + 1).

From G. C. Greubel, Apr 27 2023: (Start)

G.f.: (1 + x)*(1 + x^16)/(1 - 42*x + 861*x^16 - 820*x^17).

a(n) = 41*Sum_{k=1..15} a(n-k) - 861*a(n-16). (End)