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Expansion of 1/((1-x)(1-4x)(1-6x)(1-7x)).
1

%I #14 Sep 08 2022 08:44:45

%S 1,18,213,2098,18669,155946,1248661,9704706,73804797,552214234,

%T 4080381669,29857556274,216794571085,1564401539082,11232205936437,

%U 80315244188002,572351251736733,4067348923173690,28836875054284165

%N Expansion of 1/((1-x)(1-4x)(1-6x)(1-7x)).

%H Vincenzo Librandi, <a href="/A021804/b021804.txt">Table of n, a(n) for n = 0..200</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (18,-111,262,-168).

%F G.f.: 1/((1-x)*(1-4*x)*(1-6*x)*(1-7*x)).

%F a(n) = -1/90 +2^(2n+5)/9 -2^(n+2)*3^(n+3)/5 +7^(n+3)/18. [_Bruno Berselli_, May 08 2013]

%F a(0)=1, a(1)=18, a(2)=213, a(3)=2098, a(n)=18*a(n-1)-111*a(n-2)+ 262*a(n-3)- 168*a(n-4). - _Harvey P. Dale_, May 28 2015

%t CoefficientList[Series[1/((1 - x) (1 - 4 x) (1 - 6 x) (1 - 7 x)), {x, 0, 20}], x] (* _Bruno Berselli_, May 08 2013 *)

%t LinearRecurrence[{18,-111,262,-168},{1,18,213,2098},30] (* _Harvey P. Dale_, May 28 2015 *)

%o (PARI) Vec(1/((1-x)*(1-4*x)*(1-6*x)*(1-7*x))+O(x^20)) \\ _Bruno Berselli_, May 08 2013

%o (Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-x)*(1-4*x)*(1-6*x)*(1-7*x)))); // _Bruno Berselli_, May 08 2013

%Y Cf. A019316 (first differences).

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_.