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A016316 Expansion of 1/((1-2x)*(1-8x)*(1-9x)).

%I A016316 #44 Jan 08 2025 18:38:48
%S A016316 1,19,255,2975,32231,333759,3353335,32976175,319155111,3051352799,
%T A016316 28893830615,271497720975,2535105456391,23548956856639,
%U A016316 217804673719095,2007154559579375,18439691005140071,168959618797797279,1544655767192730775,14094055488835543375
%N A016316 Expansion of 1/((1-2x)*(1-8x)*(1-9x)).
%H A016316 Vincenzo Librandi, <a href="/A016316/b016316.txt">Table of n, a(n) for n = 0..200</a>
%H A016316 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (19,-106,144).
%F A016316 a(n) = 2^(n+1)/21 - 4*8^(n+1)/3 + 9^(n+2)/7; a(n) = A016133(n+1) - A016131(n+1). - _Zerinvary Lajos_, Jun 05 2009 [corrected by _R. J. Mathar_, Mar 14 2011]
%F A016316 From _Vincenzo Librandi_, Jun 26 2013: (Start)
%F A016316 a(n) = 19*a(n-1) - 106*a(n-2) + 144*a(n-3).
%F A016316 a(n) = 17*a(n-1) - 72*a(n-2) + 2^n. (End)
%F A016316 E.g.f.: exp(2*x)*(2 - 224*exp(6*x) + 243*exp(7*x))/21. - _Stefano Spezia_, Jul 30 2022
%t A016316 CoefficientList[Series[1 / ((1 - 2 x) (1 - 8 x) (1 - 9 x)), {x, 0, 20}], x] (* _Vincenzo Librandi_, Jun 26 2013 *)
%t A016316 LinearRecurrence[{19,-106,144},{1,19,255},30] (* _Harvey P. Dale_, Dec 29 2021 *)
%o A016316 (Sage) [(9^n - 2^n)/7-(8^n - 2^n)/6 for n in range(2,20)] # _Zerinvary Lajos_, Jun 05 2009
%o A016316 (PARI) a(n)=(9^n-2^n)/7-(8^n-2^n)/6 \\ _Charles R Greathouse IV_, Sep 24 2012
%o A016316 (Magma) I:=[1,19,255]; [n le 3 select I[n] else 19*Self(n-1)-106*Self(n-2)+144*Self(n-3): n in [1..20]]; // _Vincenzo Librandi_, Jun 26 2013
%o A016316 (Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-2*x)*(1-8*x)*(1-9*x)))); // _Vincenzo Librandi_, Jun 26 2013
%Y A016316 Cf. A016131, A016133.
%K A016316 nonn,easy
%O A016316 0,2
%A A016316 _N. J. A. Sloane_