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A019839 Expansion of 1/((1-5x)(1-6x)(1-10x)).

%I A019839 #20 Sep 08 2022 08:44:44
%S A019839 1,21,301,3681,41461,445641,4658221,47871201,486836581,4919066361,
%T A019839 49504632541,496978967121,4981629662101,49888557269481,
%U A019839 499325240101261,4995920923029441,49975372950286021,499851474762263001,4999105033876312381,49994611129771546161
%N A019839 Expansion of 1/((1-5x)(1-6x)(1-10x)).
%H A019839 Vincenzo Librandi, <a href="/A019839/b019839.txt">Table of n, a(n) for n = 0..200</a>
%H A019839 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (21,-140,300).
%F A019839 G.f.: 1/((1-5*x)*(1-6*x)*(1-10*x)).
%F A019839 a(n) = 5*5^n - 9*6^n + 5*10^n. - _R. J. Mathar_, Jun 29 2013
%F A019839 a(n) = 21*a(n-1) - 140*a(n-2) + 300*a(n-3), with a(0)=1, a(1)=21, a(2)=301. - _Vincenzo Librandi_, Jul 03 2013
%F A019839 a(n) = 16*a(n-1) - 60*a(n-2) + 5^n. - _Vincenzo Librandi_, Jul 03 2013
%p A019839 a:= n-> (Matrix(3, (i, j)-> `if`(i=j-1, 1, `if`(j=1, [21, -140, 300][i], 0)))^n)[1, 1]: seq(a(n), n=0..25);  # _Alois P. Heinz_, Jul 03 2013
%t A019839 CoefficientList[Series[1 / ((1 - 5 x) (1 - 6 x) (1 - 10 x)), {x, 0, 20}], x] (* _Vincenzo Librandi_, Jul 03 2013 *)
%o A019839 (Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-5*x)*(1-6*x)*(1-10*x)))); /* or */ I:=[1, 21, 301]; [n le 3 select I[n] else 21*Self(n-1)-140*Self(n-2)+300*Self(n-3): n in [1..20]]; // _Vincenzo Librandi_, Jul 03 2013
%K A019839 nonn,easy
%O A019839 0,2
%A A019839 _N. J. A. Sloane_