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Expansion of 1/((1-4x)(1-5x)(1-7x)).
1

%I #16 Sep 08 2022 08:44:44

%S 1,16,173,1580,13161,103656,787333,5836420,42545921,306538496,

%T 2190403293,15560186460,110074899481,776359376536,5463959472053,

%U 38396009227700,269517824177841,1890370747033776,13251393837657613

%N Expansion of 1/((1-4x)(1-5x)(1-7x)).

%H Vincenzo Librandi, <a href="/A018209/b018209.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 (16,-83,140).

%F a(n) = 16*4^n/3 -25*5^n/2 +49*7^n/6. - _R. J. Mathar_, Jun 29 2013

%F a(0)=1, a(1)=16, a(2)=173; for n>2, a(n) = 16*a(n-1) -83*a(n-2) +140*a(n-3). - _Vincenzo Librandi_, Jul 02 2013

%F a(n) = 12*a(n-1) -35*a(n-2) + 4^n. - _Vincenzo Librandi_, Jul 02 2013

%t CoefficientList[Series[1 / ((1 - 4 x) (1 - 5 x) (1 - 7 x)), {x, 0, 20}], x] (* _Vincenzo Librandi_, Jul 02 2013 *)

%t LinearRecurrence[{16,-83,140},{1,16,173},30] (* _Harvey P. Dale_, Oct 18 2013 *)

%o (Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-4*x)*(1-5*x)*(1-7*x)))); /* or */ I:=[1, 16, 173]; [n le 3 select I[n] else 16*Self(n-1)-83*Self(n-2)+140*Self(n-3): n in [1..20]]; // _Vincenzo Librandi_, Jul 02 2013

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_.