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A019333 Expansion of g.f. 1/((1-4*x)*(1-6*x)*(1-8*x)).

%I A019333 #41 May 04 2025 14:43:04
%S A019333 1,18,220,2280,21616,194208,1685440,14290560,119232256,983566848,
%T A019333 8047836160,65462691840,530198327296,4280634482688,34479631482880,
%U A019333 277245459333120,2226418414452736,17862092934217728,143201285904793600,1147437816702566400,9190468809917464576
%N A019333 Expansion of g.f. 1/((1-4*x)*(1-6*x)*(1-8*x)).
%H A019333 Vincenzo Librandi, <a href="/A019333/b019333.txt">Table of n, a(n) for n = 0..200</a>
%H A019333 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (18,-104,192).
%F A019333 a(n) = 2*4^n -9*6^n +8*8^n. - _R. J. Mathar_, Jun 29 2013
%F A019333 From _Vincenzo Librandi_, Jul 02 2013: (Start)
%F A019333 a(n) = 18*a(n-1) - 104*a(n-2) + 192*a(n-3) for n > 2.
%F A019333 a(n) = 14*a(n-1) - 48*a(n-2) + 4^n. (End)
%F A019333 E.g.f.: exp(4*x)*(2 - 9*exp(2*x) + 8*exp(4*x)). - _Stefano Spezia_, Jun 04 2024
%F A019333 From _Seiichi Manyama_, May 04 2025: (Start)
%F A019333 a(n) = Sum_{k=0..n} 2^k * 4^(n-k) * binomial(n+2,k+2) * Stirling2(k+2,2).
%F A019333 a(n) = Sum_{k=0..n} (-2)^k * 8^(n-k) * binomial(n+2,k+2) * Stirling2(k+2,2). (End)
%t A019333 CoefficientList[Series[1 / ((1 - 4 x) (1 - 6 x) (1 - 8 x)), {x, 0, 20}], x] (* _Vincenzo Librandi_, Jul 02 2013 *)
%o A019333 (PARI) Vec(1/((1-4*x)*(1-6*x)*(1-8*x))+O(x^99)) \\ _Charles R Greathouse IV_, Sep 26 2012
%o A019333 (Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-4*x)*(1-6*x)*(1-8*x)))); // _Vincenzo Librandi_, Jul 02 2013
%o A019333 (Magma) I:=[1, 18, 220]; [n le 3 select I[n] else 18*Self(n-1)-104*Self(n-2)+192*Self(n-3): n in [1..20]]; // _Vincenzo Librandi_, Jul 02 2013
%Y A019333 Equals 2^n * A016269.
%Y A019333 Cf. A017389, A019943.
%K A019333 nonn,easy
%O A019333 0,2
%A A019333 _N. J. A. Sloane_