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A163965 Number of reduced words of length n in Coxeter group on 17 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

%I A163965 #16 Sep 08 2022 08:45:47
%S A163965 1,17,272,4352,69632,1114112,17825656,285208320,4563298440,
%T A163965 73012220160,1168186644480,18690844262400,299051235428280,
%U A163965 4784783402808000,76555952624637000,1224885932940283200,19598025983313945600
%N A163965 Number of reduced words of length n in Coxeter group on 17 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.
%C A163965 The initial terms coincide with those of A170736, although the two sequences are eventually different.
%C A163965 Computed with MAGMA using commands similar to those used to compute A154638.
%H A163965 G. C. Greubel, <a href="/A163965/b163965.txt">Table of n, a(n) for n = 0..825</a>
%H A163965 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (15,15,15,15,15,-120).
%F A163965 G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(120*t^6 - 15*t^5 - 15*t^4 - 15*t^3 - 15*t^2 - 15*t + 1).
%F A163965 a(n) = -120*a(n-6) + 15*Sum_{k=1..5} a(n-k). - _Wesley Ivan Hurt_, May 11 2021
%p A163965 seq(coeff(series((1+t)*(1-t^6)/(1-16*t+135*t^6-120*t^7), t, n+1), t, n), n = 0 .. 30); # _G. C. Greubel_, Aug 11 2019
%t A163965 CoefficientList[Series[(1+t)*(1-t^6)/(1-16*t+135*t^6-120*t^7), {t,0,30}], t] (* _G. C. Greubel_, Aug 23 2017 *)
%t A163965 coxG[{6, 120, -15}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Aug 11 2019 *)
%o A163965 (PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-16*t+135*t^6-120*t^7)) \\ _G. C. Greubel_, Aug 23 2017
%o A163965 (Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-16*t+135*t^6-120*t^7) )); // _G. C. Greubel_, Aug 11 2019
%o A163965 (Sage)
%o A163965 def A163965_list(prec):
%o A163965     P.<t> = PowerSeriesRing(ZZ, prec)
%o A163965     return P((1+t)*(1-t^6)/(1-16*t+135*t^6-120*t^7)).list()
%o A163965 A163965_list(30) # _G. C. Greubel_, Aug 11 2019
%o A163965 (GAP) a:=[17, 272, 4352, 69632, 1114112, 17825656];; for n in [7..30] do a[n]:=15*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -120*a[n-6]; od; Concatenation([1], a); # _G. C. Greubel_, Aug 11 2019
%K A163965 nonn
%O A163965 0,2
%A A163965 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009