A167922 - OEIS

A167922

Number of reduced words of length n in Coxeter group on 14 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

1, 14, 182, 2366, 30758, 399854, 5198102, 67575326, 878479238, 11420230094, 148462991222, 1930018885886, 25090245516518, 326173191714734, 4240251492291542, 55123269399790046, 716602502197270507

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OFFSET

0,2

COMMENTS

The initial terms coincide with those of A170733, 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 (12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,-78).

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)/( 78*t^16 - 12*t^15 - 12*t^14 - 12*t^13 - 12*t^12 - 12*t^11 - 12*t^10 - 12*t^9 - 12*t^8 - 12*t^7 - 12*t^6 - 12*t^5 - 12*t^4 - 12*t^3 - 12*t^2 - 12*t + 1).

From G. C. Greubel, Sep 13 2023: (Start)

G.f.: (1+t)*(1-t^16)/(1 - 13*t + 90*t^16 - 78*t^17).

a(n) = 12*Sum_{j=1..15} a(n-j) - 78*a(n-16). (End)

MATHEMATICA

CoefficientList[Series[(1+t)*(1-t^16)/(1-13*t+90*t^16-78*t^17), {t, 0, 50}], t] (* G. C. Greubel, Jul 01 2016; Sep 13 2023 *)

coxG[{16, 78, -12, 40}] (* The coxG program is at A169452 *) (* G. C. Greubel, Sep 13 2023 *)

PROG

(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-13*x+90*x^16-78*x^17) )); // G. C. Greubel, Sep 13 2023

(SageMath)

def A167922_list(prec):

P.<x> = PowerSeriesRing(ZZ, prec)

return P( (1+x)*(1-x^16)/(1-13*x+90*x^16-78*x^17) ).list()