A167957 - OEIS

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

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OFFSET

0,2

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).

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)

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 *)

PROG

(Magma) R<x>:=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

(SageMath)

def A167957_list(prec):

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

return P( (1+x)*(1-x^16)/(1-40*x+819*x^16-780*x^17) ).list()