%I #16 Sep 13 2023 08:49:42

%S 1,14,182,2366,30758,399854,5198102,67575326,878479238,11420230094,

%T 148462991222,1930018885886,25090245516518,326173191714734,

%U 4240251492291542,55123269399790046,716602502197270507

%N Number of reduced words of length n in Coxeter group on 14 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

%C The initial terms coincide with those of A170733, 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="/A167922/b167922.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 (12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,-78).

%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)/( 78*t^16 - 12*t^15 - 12*t^14 - 12*t^13 - 12*t^12 - 12*t^11 - 12*t^10 - 12*t^9 - 12*t^8 - 12*t^7 - 12*t^6 - 12*t^5 - 12*t^4 - 12*t^3 - 12*t^2 - 12*t + 1).

%F From _G. C. Greubel_, Sep 13 2023: (Start)

%F G.f.: (1+t)*(1-t^16)/(1 - 13*t + 90*t^16 - 78*t^17).

%F a(n) = 12*Sum_{j=1..15} a(n-j) - 78*a(n-16). (End)

%t CoefficientList[Series[(1+t)*(1-t^16)/(1-13*t+90*t^16-78*t^17), {t, 0, 50}], t] (* _G. C. Greubel_, Jul 01 2016; Sep 13 2023 *)

%t coxG[{16, 78, -12, 40}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Sep 13 2023 *)

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-13*x+90*x^16-78*x^17) )); // _G. C. Greubel_, Sep 13 2023

%o (SageMath)

%o def A167922_list(prec):

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

%o return P( (1+x)*(1-x^16)/(1-13*x+90*x^16-78*x^17) ).list()

%o A167922_list(40) # _G. C. Greubel_, Sep 13 2023

%Y Cf. A154638, A169452, A170733.

%Y Cf. A167881, A167882, A167896 - A167900, A167908, A167914, A167916, A167919, A167923, A167924, A167926, A167927, A167929, A167931, A167933, A167935, A167937, A167938, A167940 - A167947, A167949 - A167962, A167978, A167980, A167988, A167989.

%K nonn

%O 0,2

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