A009117 Expansion of e.g.f.: 1/2 + exp(-4*x)/2.
1, -2, 8, -32, 128, -512, 2048, -8192, 32768, -131072, 524288, -2097152, 8388608, -33554432, 134217728, -536870912, 2147483648, -8589934592, 34359738368, -137438953472, 549755813888, -2199023255552, 8796093022208, -35184372088832, 140737488355328, -562949953421312
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
- Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.
- Katarzyna Grygiel, Pawel M. Idziak and Marek Zaionc, How big is BCI fragment of BCK logic, arXiv preprint arXiv:1112.0643 [cs.LO], 2011. - _N. J. A. Sloane_, Feb 21 2012
- INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 171
- Index entries for linear recurrences with constant coefficients, signature (-4).
Crossrefs
a(n) = (-1)^n * A004171(n-1).
Programs
-
Magma
m:=30; R
:=PowerSeriesRing(Integers(), m); Coefficients(R!((2*x+1)/(1+4*x))); // G. C. Greubel, Jul 26 2018 -
Maple
A009117:=n->`if`(n=0, 1, (-4)^n/2); seq(A009117(n), n=0..30); # Wesley Ivan Hurt, Mar 10 2014
-
Mathematica
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 *) -
PARI
x='x+O('x^100); Vec((1+2*x)/(1+4*x)) \\ Altug Alkan, Dec 21 2015
Formula
1 followed by (-4)^n /2.
E.g.f.: cos(x)^2 (even powers).
a(n) = Sum_{k, 0<=k<=n} A086872(n,k)*(-3)^(n-k). - Philippe Deléham, Aug 17 2007
G.f. (2*x+1)/(1+4*x). - R. J. Mathar, Mar 08 2011
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
a(n) = (-1)^n * A081294(n). - Philippe Deléham, Mar 09 2014
Extensions
Signs added and formula corrected by Olivier Gérard, Mar 15 1997
More terms from Olaf Voß, Feb 13 2008
Definition corrected by Joerg Arndt, May 16 2011