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A164350 Number of reduced words of length n in Coxeter group on 49 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

%I A164350 #17 Sep 08 2022 08:45:47
%S A164350 1,49,2352,112896,5419008,260112384,12485393256,599298819840,
%T A164350 28766340643992,1380784220911872,66277636363782144,
%U A164350 3181326245942132736,152703645428292064680,7329774290465429385408,351829132817899422588504,16887796785286144959221568
%N A164350 Number of reduced words of length n in Coxeter group on 49 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.
%C A164350 The initial terms coincide with those of A170768, although the two sequences are eventually different.
%C A164350 Computed with MAGMA using commands similar to those used to compute A154638.
%H A164350 G. C. Greubel, <a href="/A164350/b164350.txt">Table of n, a(n) for n = 0..590</a>
%H A164350 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (47, 47, 47, 47, 47, -1128).
%F A164350 G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(1128*t^6 - 47*t^5 - 47*t^4 - 47*t^3 - 47*t^2 - 47*t + 1).
%F A164350 a(n) = -1128*a(n-6) + 47*Sum_{k=1..5} a(n-k). - _Wesley Ivan Hurt_, May 11 2021
%p A164350 seq(coeff(series((1+t)*(1-t^6)/(1-48*t+1175*t^6-1128*t^7), t, n+1), t, n), n = 0..20); # _G. C. Greubel_, Aug 24 2019
%t A164350 CoefficientList[Series[(1+t)*(1-t^6)/(1-48*t+1175*t^6-1128*t^7), {t, 0, 20}], t] (* _G. C. Greubel_, Sep 15 2017 *)
%t A164350 coxG[{6, 1128, -47}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Aug 24 2019 *)
%o A164350 (PARI) my(t='t+O('t^20)); Vec((1+t)*(1-t^6)/(1-48*t+1175*t^6-1128*t^7)) \\ _G. C. Greubel_, Sep 15 2017
%o A164350 (Magma) R<t>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+t)*(1-t^6)/(1-48*t+1175*t^6-1128*t^7) )); // _G. C. Greubel_, Aug 24 2019
%o A164350 (Sage)
%o A164350 def A164350_list(prec):
%o A164350     P.<t> = PowerSeriesRing(ZZ, prec)
%o A164350     return P((1+t)*(1-t^6)/(1-48*t+1175*t^6-1128*t^7)).list()
%o A164350 A164350_list(20) # _G. C. Greubel_, Aug 24 2019
%o A164350 (GAP) a:=[49, 2352, 112896, 5419008, 260112384, 12485393256];; for n in [7..20] do a[n]:=47*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -1128*a[n-6]; od; Concatenation([1], a); # _G. C. Greubel_, Aug 24 2019
%K A164350 nonn
%O A164350 0,2
%A A164350 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009