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Number of reduced words of length n in Coxeter group on 34 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.
1

%I #17 Sep 06 2023 18:22:33

%S 1,34,1122,37026,1221858,40321314,1330603362,43909910946,

%T 1449027061218,47817893020194,1577990469666402,52073685498991266,

%U 1718431621466711778,56708243508401488674,1871372035777249126242,61755277180649221165986

%N Number of reduced words of length n in Coxeter group on 34 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

%C The initial terms coincide with those of A170753, although the two sequences are eventually different.

%C Computed with MAGMA using commands similar to those used to compute A154638.

%H G. C. Greubel, <a href="/A167950/b167950.txt">Table of n, a(n) for n = 0..500</a>

%H <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,-528).

%F 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)/( 528*t^16 - 32*t^15 - 32*t^14 - 32*t^13 - 32*t^12 - 32*t^11 - 32*t^10 - 32*t^9 - 32*t^8 - 32*t^7 - 32*t^6 - 32*t^5 - 32*t^4 - 32*t^3 - 32*t^2 - 32*t + 1).

%F From _G. C. Greubel_, Sep 06 2023: (Start)

%F G.f.: (1+t)*(1-t^16)/(1 - 33*t + 560*t^16 - 528*t^17).

%F a(n) = 32*Sum_{j=1..15} a(n-j) - 528*a(n-16). (End)

%t coxG[{16,528,-32}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, May 24 2015 *)

%t CoefficientList[Series[(1+t)*(1-t^16)/(1-33*t+560*t^16-528*t^17), {t, 0, 50}], t] (* _G. C. Greubel_, Jul 02 2016; Sep 06 2023 *)

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-33*x+560*x^16-528*x^17) )); // _G. C. Greubel_, Sep 06 2023

%o (SageMath)

%o def A167955_list(prec):

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

%o return P( (1+x)*(1-x^16)/(1-33*x+560*x^16-528*x^17) ).list()

%o A167955_list(40) # _G. C. Greubel_, Sep 06 2023

%Y Cf. A154638, A169452, A170753.

%K nonn

%O 0,2

%A _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009