A226804 - OEIS

%I #26 Feb 03 2025 18:54:18

%S 1,120,10890,914760,74987451,6098153040,494603769780,40080553611120,

%T 3247001410058901,263019982119962760,21304965990387308670,

%U 1725711626112542281080,139782894999601166334351,11322421333652608793333280,917116312670388314093059560

%N Expansion of 1/((1-3x)(1-9x)(1-27x)(1-81x)).

%H Vincenzo Librandi, <a href="/A226804/b226804.txt">Table of n, a(n) for n = 0..200</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (120,-3510,29160,-59049).

%F G.f.: 1/((1-3*x)*(1-9*x)*(1-27*x)*(1-81*x)).

%F a(n) = 3^n*(3^(n+1)-1)*(3^(n+2)-1)*(3^(n+3)-1)/416.

%F a(0)=1, a(1)=120, a(2)=10890, a(3)=914760; for n>3, a(n) = 120*a(n-1) -3510*a(n-2) +29160*a(n-3) -59049*a(n-4).

%F a(n) -108*a(n-1) +2187*a(n-2) = A016142(n) with a(-1)=a(-2)=0. [_Bruno Berselli_, Jul 11 2013]

%t CoefficientList[Series[1 / ((1 - 3 x) (1 - 9 x) (1 - 27 x) (1 - 81 x)), {x, 0, 20}], x]

%t LinearRecurrence[{120,-3510,29160,-59049},{1,120,10890,914760},20] (* _Harvey P. Dale_, Sep 21 2016 *)

%o (Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-3*x)*(1-9*x)*(1-27*x)*(1-81*x))));

%o (Magma) I:=[1,120,10890,914760]; [n le 4 select I[n] else 120*Self(n-1)-3510*Self(n-2)+29160*Self(n-3)-59049*Self(n-4): n in [1..25]];

%Y Cf. A000292, A016142, A028258.

%K nonn,easy

%O 0,2

%A _Vincenzo Librandi_, Jul 11 2013