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A084567 Binomial transform of (1,-1,4,-16,64,-256,1024,...) = (3*0^n-(-4)^n)/4.

%I A084567 #30 Aug 26 2024 15:57:01
%S A084567 1,0,3,-6,21,-60,183,-546,1641,-4920,14763,-44286,132861,-398580,
%T A084567 1195743,-3587226,10761681,-32285040,96855123,-290565366,871696101,
%U A084567 -2615088300,7845264903,-23535794706,70607384121,-211822152360,635466457083,-1906399371246
%N A084567 Binomial transform of (1,-1,4,-16,64,-256,1024,...) = (3*0^n-(-4)^n)/4.
%C A084567 Partial sums of (1,-1,3,-9,27,-81,....) (with g.f. (1+2x)/(1+3x) ).
%H A084567 Vincenzo Librandi, <a href="/A084567/b084567.txt">Table of n, a(n) for n = 0..1000</a>
%H A084567 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-2,3)
%F A084567 G.f.: (1+2*x)/((1-x)*(1+3*x)).
%F A084567 G.f.: 1+ x -x/Q(0), where Q(k) = 1 + 3*x^2 + (3*k+4)*x - x*(3*k+1 + 3*x)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Oct 07 2013
%t A084567 CoefficientList[Series[(1 + 2 x)/((1 - x) (1 + 3 x)), {x, 0, 50}], x] (* _Vincenzo Librandi_, Oct 07 2013 *)
%t A084567 LinearRecurrence[{-2,3},{1,0},30] (* _Harvey P. Dale_, Aug 26 2024 *)
%o A084567 (PARI) Vec((1+2*x)/((1-x)*(1+3*x))+O(x^66)) \\ _Joerg Arndt_, Jul 14 2013
%o A084567 (Magma) I:=[1,0]; [n le 2 select I[n] else -2*Self(n-1)+3*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Oct 07 2013
%Y A084567 Cf. A054878 (absolute values).
%K A084567 easy,sign
%O A084567 0,3
%A A084567 _Paul Barry_, May 30 2003
%E A084567 Removed incorrect g.f. and e.g.f., _Joerg Arndt_, Jul 14 2013