A167946 Number of reduced words of length n in Coxeter group on 31 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

1, 31, 930, 27900, 837000, 25110000, 753300000, 22599000000, 677970000000, 20339100000000, 610173000000000, 18305190000000000, 549155700000000000, 16474671000000000000, 494240130000000000000, 14827203900000000000000

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170750, 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 (29,29,29,29,29,29,29,29,29,29,29,29,29,29,29,-435).

Crossrefs

Cf. A154638, A169452, A170750.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-30*x+464*x^16-435*x^17) )); // G. C. Greubel, Sep 07 2023

Mathematica

CoefficientList[Series[(1+t)*(1-t^16)/(1-30*t+464*t^16-435*t^17), {t, 0, 50}], t] (* G. C. Greubel, Jul 02 2016; Sep 07 2023 *)
coxG[{16, 435, -29, 40}] (* The coxG program is at A169452 *) (* G. C. Greubel, Sep 07 2023 *)

SageMath

def A167946_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^16)/(1-30*x+464*x^16-435*x^17) ).list()
A167946_list(40) # G. C. Greubel, Sep 07 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)/( 435*t^16 - 29*t^15 - 29*t^14 - 29*t^13 - 29*t^12 - 29*t^11 - 29*t^10 - 29*t^9 - 29*t^8 - 29*t^7 - 29*t^6 - 29*t^5 - 29*t^4 - 29*t^3 - 29*t^2 - 29*t + 1).

From G. C. Greubel, Sep 07 2023: (Start)

G.f.: (1+t)*(1-t^16)/(1 - 30*t + 464*t^16 - 435*t^17).

a(n) = 37*Sum_{j=1..15} a(n-j) - 703*a(n-16). (End)