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%I #59 Sep 08 2022 08:44:37
%S 1,-2,8,-32,128,-512,2048,-8192,32768,-131072,524288,-2097152,8388608,
%T -33554432,134217728,-536870912,2147483648,-8589934592,34359738368,
%U -137438953472,549755813888,-2199023255552,8796093022208,-35184372088832,140737488355328,-562949953421312
%N Expansion of e.g.f.: 1/2 + exp(-4*x)/2.
%H Vincenzo Librandi, <a href="/A009117/b009117.txt">Table of n, a(n) for n = 0..200</a>
%H Ghislain R. Franssens, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Franssens/franssens13.html">On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.
%H Katarzyna Grygiel, Pawel M. Idziak and Marek Zaionc, <a href="http://arxiv.org/abs/1112.0643">How big is BCI fragment of BCK logic</a>, arXiv preprint arXiv:1112.0643 [cs.LO], 2011. - _N. J. A. Sloane_, Feb 21 2012
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=171">Encyclopedia of Combinatorial Structures 171</a>
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (-4).
%F 1 followed by (-4)^n /2.
%F E.g.f.: cos(x)^2 (even powers).
%F a(n) = Sum_{k, 0<=k<=n} A086872(n,k)*(-3)^(n-k). - _Philippe Deléham_, Aug 17 2007
%F G.f. (2*x+1)/(1+4*x). - _R. J. Mathar_, Mar 08 2011
%F E.g.f.: 1/2 + exp(-4*x)/2 = (G(0)+1)/2 ; G(k) = 1 - 4*x/(2*k+1 - 2*x*(2*k+1)/(2*x - (k+1)/G(k+1))) ; (continued fraction). - _Sergei N. Gladkovskii_, Dec 20 2011
%F a(n) = (-1)^n * A081294(n). - _Philippe Deléham_, Mar 09 2014
%p A009117:=n->`if`(n=0, 1, (-4)^n/2); seq(A009117(n), n=0..30); # _Wesley Ivan Hurt_, Mar 10 2014
%t With[{nn=30},CoefficientList[Series[1/2+Exp[-4x]/2,{x,0,nn}],x] Range[ 0,nn]!] (* or *) LinearRecurrence[{-4},{1,-2},30] (* _Harvey P. Dale_, Apr 09 2015 *)
%o (PARI) x='x+O('x^100); Vec((1+2*x)/(1+4*x)) \\ _Altug Alkan_, Dec 21 2015
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((2*x+1)/(1+4*x))); // _G. C. Greubel_, Jul 26 2018
%Y a(n) = (-1)^n * A004171(n-1).
%K sign,easy
%O 0,2
%A _R. H. Hardin_
%E Signs added and formula corrected by _Olivier Gérard_, Mar 15 1997
%E More terms from _Olaf Voß_, Feb 13 2008
%E Definition corrected by _Joerg Arndt_, May 16 2011
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