%I #34 Oct 03 2025 16:34:18

%S 1,18,225,2430,24381,234738,2205225,20404710,186995061,1703091258,

%T 15448694625,139763668590,1262226050541,11386154248578,

%U 102632111782425,924629361662070,8327306431726821,74979611075290698

%N Expansion of 1/((1-3x)(1-6x)(1-9x)).

%H Vincenzo Librandi, <a href="/A017933/b017933.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 (18,-99,162).

%F a(n) = (3^n)*Stirling2(n+3, 3), n >= 0, with Stirling2(n, m) = A008277(n, m).

%F a(n) = (3^n - 8*6^n + 9*9^n)/2.

%F G.f.: 1/((1-3*x)*(1-6*x)*(1-9*x)).

%F E.g.f.: (d^3/dx^3)((((exp(3*x)-1)/3)^3)/3!) = (exp(3*x) - 8*exp(6*x) + 9*exp(9*x))/2.

%F a(0)=1, a(1)=18, a(2)=225; for n > 2, a(n) = 18*a(n-1) - 99*a(n-2) + 162*a(n-3). - _Vincenzo Librandi_, Jul 02 2013

%F a(n) = 15*a(n-1) - 54*a(n-2) + 3^n. - _Vincenzo Librandi_, Jul 02 2013

%t CoefficientList[Series[1 / ((1 - 3 x) (1 - 6 x) (1 - 9 x)), {x, 0, 30}], x] (* _Vincenzo Librandi_, Jul 02 2013 *)

%t LinearRecurrence[{18,-99,162},{1,18,225},20] (* _Harvey P. Dale_, Sep 09 2023 *)

%o (PARI) Vec(1/((1-3*x)*(1-6*x)*(1-9*x))+O(x^99)) \\ _Charles R Greathouse IV_, Sep 26 2012

%o (Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-3*x)*(1-6*x)*(1-9*x)))); // _Vincenzo Librandi_, Jul 02 2013

%o (Magma) I:=[1, 18, 225]; [n le 3 select I[n] else 18*Self(n-1)-99*Self(n-2)+162*Self(n-3): n in [1..20]]; // _Vincenzo Librandi_, Jul 02 2013

%Y Third column of triangle A075498.

%Y Cf. A016137, A028085.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_