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Expansion of 1/((1-4*x)*(1-7*x)*(1-12*x)).
2

%I #21 Sep 08 2022 08:44:44

%S 1,23,369,5143,66977,841575,10367953,126315191,1529146113,18443562247,

%T 221980457777,2668373663319,32052757927009,384859080003239,

%U 4619891122628241,55449769683406327,665474773978915265

%N Expansion of 1/((1-4*x)*(1-7*x)*(1-12*x)).

%H Vincenzo Librandi, <a href="/A019628/b019628.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 (23,-160,336).

%F a(n) = 2*4^n/3 - 7^(n+2)/15 + 18*12^n/5. - _R. J. Mathar_, Nov 11 2012

%F a(0)=1, a(1)=23, a(2)=369; for n>2, a(n) = 23*a(n-1) -160*a(n-2) +336*a(n-3). - _Vincenzo Librandi_, Jul 03 2013

%F a(n) = 19*a(n-1) - 84*a(n-2) + 4^n. - _Vincenzo Librandi_, Jul 03 2013

%t CoefficientList[Series[1 / ((1 - 4 x) (1 - 7 x) (1 - 12 x)), {x, 0, 20}], x] (* _Vincenzo Librandi_, Jul 03 2013 *)

%t LinearRecurrence[{23,-160,336}, {1,23,369}, 30] (* _G. C. Greubel_, Jan 28 2018 *)

%o (Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-4*x)*(1-7*x)*(1-12*x)))); /* or */ I:=[1, 23, 369]; [n le 3 select I[n] else 23*Self(n-1)-160*Self(n-2)+336*Self(n-3): n in [1..20]]; // _Vincenzo Librandi_, Jul 03 2013

%o (PARI) x='x+O('x^30); Vec(1/((1-4*x)*(1-7*x)*(1-12*x))) \\ _G. C. Greubel_, Jan 28 2018

%Y Cf. A021894 (partial sums).

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_