|
%I #18 Sep 08 2022 08:45:47
%S 1,50,2450,120050,5882450,288240050,14123761225,692064240000,
%T 33911144820000,1661645952120000,81420644594940000,
%U 3989611239264000000,195490933775422559400,9579054924518618851200,469373650608038610268800
%N Number of reduced words of length n in Coxeter group on 50 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.
%C The initial terms coincide with those of A170769, 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="/A164351/b164351.txt">Table of n, a(n) for n = 0..590</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (48, 48, 48, 48, 48, -1176).
%F G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(1176*t^6 - 48*t^5 - 48*t^4 - 48*t^3 - 48*t^2 - 48*t + 1).
%F a(n) = -1176*a(n-6) + 48*Sum_{k=1..5} a(n-k). - _Wesley Ivan Hurt_, May 11 2021
%p seq(coeff(series((1+t)*(1-t^6)/(1-49*t+1224*t^6-1176*t^7), t, n+1), t, n), n = 0 .. 20); # _G. C. Greubel_, Aug 24 2019
%t coxG[{6,1176,-48}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, Feb 18 2015 *)
%t CoefficientList[Series[(1+t)*(1-t^6)/(1-49*t+1224*t^6-1176*t^7), {t, 0, 20}], t] (* _G. C. Greubel_, Sep 15 2017 *)
%o (PARI) my(t='t+O('t^20)); Vec((1+t)*(1-t^6)/(1-49*t+1224*t^6-1176*t^7)) \\ _G. C. Greubel_, Sep 15 2017
%o (Magma) R<t>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+t)*(1-t^6)/(1-49*t+1224*t^6-1176*t^7) )); // _G. C. Greubel_, Aug 24 2019
%o (Sage)
%o def A164351_list(prec):
%o P.<t> = PowerSeriesRing(ZZ, prec)
%o return P((1+t)*(1-t^6)/(1-49*t+1224*t^6-1176*t^7)).list()
%o A164351_list(20) # _G. C. Greubel_, Aug 24 2019
%o (GAP) a:=[50, 2450, 120050, 5882450, 288240050, 14123761225];; for n in [7..20] do a[n]:=48*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -1176*a[n-6]; od; Concatenation([1], a); # _G. C. Greubel_, Aug 24 2019
%K nonn
%O 0,2
%A _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009
| 