A167957 - OEIS

%I #19 Jul 14 2023 06:31:15

%S 1,41,1640,65600,2624000,104960000,4198400000,167936000000,

%T 6717440000000,268697600000000,10747904000000000,429916160000000000,

%U 17196646400000000000,687865856000000000000,27514634240000000000000,1100585369600000000000000,44023414783999999999999180

%N Number of reduced words of length n in Coxeter group on 41 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

%C The initial terms coincide with those of A170760, 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="/A167957/b167957.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 (39,39,39,39,39,39,39,39,39,39,39,39,39,39,39,-780).

%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)/( 780*t^16 - 39*t^15 - 39*t^14 - 39*t^13 - 39*t^12 - 39*t^11 - 39*t^10 - 39*t^9 - 39*t^8 - 39*t^7 - 39*t^6 - 39*t^5 - 39*t^4 - 39*t^3 - 39*t^2 - 39*t + 1).

%F From _G. C. Greubel_, Jul 14 2023: (Start)

%F G.f.: (1+t)*(1-t^16)/(1 -40*t +819*t^16 -780*t^17).

%F a(n) = -780*a(n-16) + 39*Sum_{j=1..15} a(n-j). (End)

%t CoefficientList[Series[(1+t)*(1-t^16)/(1-40*t+819*t^16-780*t^17), {t, 0, 40}], t] (* _G. C. Greubel_, Jul 02 2016; Jul 14 2023 *)

%t coxG[{16,780,-39}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, Nov 20 2021 *)

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-40*x+819*x^16-780*x^17) )); // _G. C. Greubel_, Jul 14 2023

%o (SageMath)

%o def A167957_list(prec):

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

%o     return P( (1+x)*(1-x^16)/(1-40*x+819*x^16-780*x^17) ).list()

%o A167957_list(40) # _G. C. Greubel_, Jul 14 2023

%Y Cf. A154638, A169452, A170760.

%K nonn

%O 0,2

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