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

Offset: 0

Author

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

Keywords

• nonn

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).

Crossrefs

Cf. A154638, A169452, A170749.

Programs

Magma

R:=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

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 *)

SageMath

def A167945_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^16)/(1-29*x+434*x^16-406*x^17) ).list()
A167945_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)/( 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)