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

0,2

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).

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

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 *)

Prog

(PARI) x='x+O('x^100); Vec((1+2*x)/(1+4*x)) \\ Altug Alkan, Dec 21 2015
(Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((2*x+1)/(1+4*x))); // G. C. Greubel, Jul 26 2018

Crossrefs

a(n) = (-1)^n * A004171(n-1).

Sequence in context: A274524 A081294 A004171 * A331407 A160637 A183895

Adjacent sequences: A009114 A009115 A009116 * A009118 A009119 A009120

Keyword

sign,easy

Author

R. H. Hardin

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

Status

approved