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Number of reduced words of length n in Coxeter group on 33 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.
7

%I #20 Sep 07 2023 18:20:03

%S 1,33,1056,33792,1081344,34603008,1107296256,35433480192,

%T 1133871366144,36283883716608,1161084278931456,37154696925806592,

%U 1188950301625810944,38046409652025950208,1217485108864830406656,38959523483674573012992

%N Number of reduced words of length n in Coxeter group on 33 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

%C The initial terms coincide with those of A170752, 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="/A167949/b167949.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 (31,31,31,31,31,31,31,31,31,31,31,31,31,31,31,-496).

%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)/( 496*t^16 - 31*t^15 - 31*t^14 - 31*t^13 - 31*t^12 - 31*t^11 - 31*t^10 - 31*t^9 - 31*t^8 - 31*t^7 - 31*t^6 - 31*t^5 - 31*t^4 - 31*t^3 - 31*t^2 - 31*t + 1).

%F From _G. C. Greubel_, Sep 07 2023: (Start)

%F G.f.: (1+t)*(1-t^16)/(1 - 32*t + 527*t^16 - 496*t^17).

%F a(n) = 31*Sum_{j=1..15} a(n-j) - 496*a(n-16). (End)

%t CoefficientList[Series[(1+t)*(1-t^16)/(1-32*t+527*t^16-496*t^17), {t, 0, 50}], t] (* _G. C. Greubel_, Jul 02 2016; Sep 07 2023 *)

%t coxG[{16,496,-31}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, Sep 22 2020 *)

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-32*x+527*x^16-496*x^17) )); // _G. C. Greubel_, Sep 07 2023

%o (SageMath)

%o def A167949_list(prec):

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

%o return P( (1+x)*(1-x^16)/(1-32*x+527*x^16-496*x^17) ).list()

%o A167949_list(40) # _G. C. Greubel_, Sep 07 2023

%Y Cf. A154638, A169452, A170752.

%K nonn

%O 0,2

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