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%I #11 Sep 08 2022 08:46:05
%S 1,39,1170,32670,891891,24169509,653373540,17648258940,476567558181,
%T 12867905191779,347438670325110,9380891170278810,253284485241566871,
%U 6838684914320250849,184644527001833063880,4985402537886183692280,134605871302457221445961
%N Expansion of 1/((1-3x)(1-9x)(1-27x)).
%H Vincenzo Librandi, <a href="/A227524/b227524.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 (39,-351,729).
%F G.f.: 1/((1-3*x)*(1-9*x)*(1-27*x)).
%F a(n) = 3^n*(3^(n+1)-1)*(3^(n+2)-1)/16.
%F a(0)=1, a(1)=39, a(2)=1170; for n>2, a(n) = 39*a(n-1)-351*a(n-2)+729*a(n-3).
%F a(n)-27*a(n-1) = A016142(n), with a(-1)=0; a(n) = A226804(n)-81*A226804(n-1), with A226804(-1)=0. [_Bruno Berselli_, Jul 17 2013]
%t CoefficientList[Series[1 / ((1 - 3 x) (1 - 9 x) (1 - 27 x)), {x, 0, 30}], x]
%t LinearRecurrence[{39,-351,729},{1,39,1170},20] (* _Harvey P. Dale_, Jul 04 2022 *)
%o (Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-3*x)*(1-9*x)*(1-27*x)))); /* or */ I:=[1, 39, 1170]; [n le 3 select I[n] else 39*Self(n-1)-351*Self(n-2)+729*Self(n-3): n in [1..25]];
%Y Cf. A016142, A226804.
%K nonn,easy
%O 0,2
%A _Vincenzo Librandi_, Jul 17 2013
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