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%I #21 Sep 30 2023 09:18:22
%S 1,21,313,4125,51601,630741,7630633,91892685,1104403201,13261555461,
%T 159183299353,1910426955645,22926277062001,275121159824181,
%U 3301483361726473,39617948633641005,475416129363276001
%N Expansion of 1/((1-4x)(1-5x)(1-12x)).
%H Vincenzo Librandi, <a href="/A019041/b019041.txt">Table of n, a(n) for n = 0..200</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (21,-128,240).
%F a(n) = 2*4^n - 25*5^n/7 + 18*12^n/7. - _R. J. Mathar_, Jun 29 2013
%F From _Vincenzo Librandi_, Jul 02 2013: (Start)
%F a(n) = 21*a(n-1) - 128*a(n-2) + 240*a(n-3) for n > 2; a(0)=1, a(1)=21, a(2)=313.
%F a(n) = 17*a(n-1) - 60*a(n-2) + 4^n. (End)
%t CoefficientList[Series[1 / ((1 - 4 x) (1 - 5 x) (1 - 12 x)), {x, 0, 20}], x] (* _Vincenzo Librandi_, Jul 02 2013 *)
%t LinearRecurrence[{21,-128,240},{1,21,313},20] (* _Harvey P. Dale_, Mar 09 2022 *)
%o (Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-4*x)*(1-5*x)*(1-12*x)))); /* or */ I:=[1, 21, 313]; [n le 3 select I[n] else 21*Self(n-1)-128*Self(n-2)+240*Self(n-3): n in [1..20]]; // _Vincenzo Librandi_, Jul 02 2013
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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