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Expansion of 1/((1-3x)(1-7x)(1-10x)).
1

%I #31 Sep 08 2022 08:44:44

%S 1,20,279,3370,37841,407640,4281739,44256950,452652981,4597133860,

%T 46465625999,468116448930,4705386343321,47223418005680,

%U 473421066847059,4742518890351310,47483346499724861,475240568299871100,4755255406504488919,47572502130747596090

%N Expansion of 1/((1-3x)(1-7x)(1-10x)).

%H Vincenzo Librandi, <a href="/A017999/b017999.txt">Table of n, a(n) for n = 0..200</a>

%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (20,-121,210).

%F a(n) = (4*10^(n+2) - 7^(n+3) + 3^(n+3))/84. - _Yahia Kahloune_, Jun 30 2013

%F a(0)=1, a(1)=20, a(2)=279; for n>2, a(n) = 20*a(n-1) -121*a(n-2) +210*a(n-3). - _Vincenzo Librandi_, Jul 02 2013

%F a(n) = 17*a(n-1) -70*a(n-2) +3^n. - _Vincenzo Librandi_, Jul 02 2013

%p a:= n-> (Matrix(3, (i, j)-> `if`(i=j-1, 1, `if`(j=1, [20, -121, 210][i], 0)))^n)[1, 1]: seq(a(n), n=0..25); # _Alois P. Heinz_, Jul 02 2013

%t CoefficientList[Series[1 / ((1 - 3 x) (1 - 7 x) (1 - 10 x)), {x, 0, 30}], x] (* _Vincenzo Librandi_, Jul 02 2013 *)

%o (Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-3*x)*(1-7*x)*(1-10*x)))); /* or */ I:=[1, 20, 279]; [n le 3 select I[n] else 20*Self(n-1)-121*Self(n-2)+210*Self(n-3): n in [1..20]]; // _Vincenzo Librandi_, Jul 02 2013

%Y First differences of A021604.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_.