A167945 - OEIS

A167945

Number of reduced words of length n in Coxeter group on 30 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

1, 30, 870, 25230, 731670, 21218430, 615334470, 17844699630, 517496289270, 15007392388830, 435214379276070, 12621216999006030, 366015292971174870, 10614443496164071230, 307818861388758065670, 8926746980273983904430

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OFFSET

0,2

COMMENTS

The initial terms coincide with those of A170749, 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 (28,28,28,28,28,28,28,28,28,28,28,28,28,28,28,-406).

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)/( 406*t^16 - 28*t^15 - 28*t^14 - 28*t^13 - 28*t^12 - 28*t^11 - 28*t^10 - 28*t^9 - 28*t^8 - 28*t^7 - 28*t^6 - 28*t^5 - 28*t^4 - 28*t^3 - 28*t^2 - 28*t + 1).

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

G.f.: (1+t)*(1-t^16)/(1 - 29*t + 434*t^16 - 406*t^17).

a(n) = 28*Sum_{j=1..15} a(n-j) - 406*a(n-16). (End)

MATHEMATICA

CoefficientList[Series[(1+t)*(1-t^16)/(1-29*t+434*t^16-406*t^17), {t, 0, 50}], t] (* G. C. Greubel, Jul 02 2016; Sep 07 2023 *)

coxG[{16, 406, -28}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Dec 15 2017 *)

PROG

(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-29*x+434*x^16-406*x^17) )); // G. C. Greubel, Sep 07 2023

(SageMath)

def A167945_list(prec):

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

return P( (1+x)*(1-x^16)/(1-29*x+434*x^16-406*x^17) ).list()