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Expansion of 1/((1-3*x)*(1-7*x)*(1-11*x)).
3

%I #32 Sep 08 2022 08:44:44

%S 1,21,310,3990,48031,557571,6338620,71164380,792891661,8792412321,

%T 97210822930,1072779241170,11824793506891,130242283148271,

%U 1433852001421240,15780680237514360,173645640208869721,1910509145600189421,21018450325107861550,231222901641889183950

%N Expansion of 1/((1-3*x)*(1-7*x)*(1-11*x)).

%H Vincenzo Librandi, <a href="/A018054/b018054.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 (21,-131,231).

%F a(0)=1, a(1)=21, a(2)=310; for n > 2, a(n) = 21*a(n-1) - 131*a(n-2) + 231*a(n-3). - _Vincenzo Librandi_, Jul 02 2013

%F a(n) = 18*a(n-1) - 77*a(n-2) + 3^n. - _Vincenzo Librandi_, Jul 02 2013

%F a(n) = (11^(n+2) - 2*7^(n+2) + 3^(n+2))/32. - _Yahia Kahloune_, Jul 06 2013

%F O.g.f.: 1/((1-3*x) * (1-7*x) * (1-11*x)).

%F E.g.f.: (d^2/dx^2)(exp(3*x)*(exp(4*x)-1)^2/(4^2*2!)) = exp(3*x)*(121*exp(8*x) - 98*exp(4*x) + 9)/32. - _Wolfdieter Lang_, Apr 13 2017

%p a:= n-> (Matrix(3, (i, j)-> `if`(i=j-1, 1, `if`(j=1, [21, -131, 231][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 - 11 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)*(1-11*x)))); /* or */ I:=[1, 21, 310]; [n le 3 select I[n] else 21*Self(n-1)-131*Self(n-2)+231*Self(n-3): n in [1..20]]; // _Vincenzo Librandi_, Jul 02 2013

%Y Cf. Column k=2 of A225469.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_