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A009116
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E.g.f. cos(x) / exp(x).
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28
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1, -1, 0, 2, -4, 4, 0, -8, 16, -16, 0, 32, -64, 64, 0, -128, 256, -256, 0, 512, -1024, 1024, 0, -2048, 4096, -4096, 0, 8192, -16384, 16384, 0, -32768, 65536, -65536, 0, 131072, -262144, 262144, 0, -524288, 1048576, -1048576, 0, 2097152, -4194304
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Apart from signs, generated by 1,1 position of H_2^n=[1,1;-1,1]^n; and a(n)=2^(n/2)*cos(pi*n/2) - Paul Barry, Feb 18 2004
Equals binomial transform of "Period 4, repeat [1, 0, -1, 0]". [From Gary W. Adamson, Mar 25 2009]
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LINKS
| Index entries for sequences related to linear recurrences with constant coefficients, signature (-2,-2).
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FORMULA
| Real part of (-1-i)^n - Marc LeBrun (mlb(AT)well.com)
a(n) = -2 * (a(n-1) + a(n-2)), a(0) = 1, a(1) = -1 - Michael Somos, Nov 17 2002
Sum_{j=0..[n/2]} (-1)^j*binomial(n, 2*j).
G.f.: (1 + x) / (1 + 2*x + 2*x^2).
E.g.f.: cos(x) / exp(x).
a(n)= Sum_{k, 0<=k<=n}(-1)^k*A098158(n,k) . - Philippe DELEHAM, Dec 04 2006
a(n)*(-1)^n=A099087(n)-A099087(n-1). - R. J. Mathar, Nov 18 2007
a(n)=1/2*[(-1-I)^n+(-1+I)^n], with n>=0 and I=sqrt(-1) [From Paolo P. Lava, Nov 21 2008]
a(n)=(-1)^n*A146559(n). [From Philippe DELEHAM, Dec 01 2008]
a(n)=-4*a(n-4); a(n)=A016116(n) * A075553(n+6). - Paul Curtz, Jul 22 2011
E.g.f.: cos(x)/exp(x) =1-x/(G(0)+1); G(k)=4k+1-x+(x^2)*(4k+1)/((2k+1)*(4k+3)-(x^2)+x*(2k+1)*(4k+3)/( 2k+2-x+x*(2k+2)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 24 2011
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EXAMPLE
| 1 - x + 2*x^3 - 4*x^4 + 4*x^5 - 8*x^7 + 16*x^8 - 16*x^9 + 32*x^11 - 64*x^12 + ...
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MAPLE
| A009116 := n->add((-1)^j*binomial(n, 2*j), j=0..floor(n/2));
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MATHEMATICA
| n = 50; (* n = 2 mod 4 *) (CoefficientList[ Series[ Cos[x]/Exp[x], {x, 0, n}], x]* Table[k!, {k, 0, n - 1}] )[[1 ;; 45]] (* From Jean-François Alcover, May 18 2011 *)
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PROG
| (PARI) {a(n) = if( n<0, 0, polcoeff( (1 + x) / (1 + 2*x + 2*x^2) + x * O(x^n), n))} /* Michael Somos Nov 17 2002 */
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CROSSREFS
| Cf. A009545.
Cf. A090132.
(With different signs) Row sums of triangle A104597.
Sequence in context: A195479 A112793 * A146559 A118434 A090132 A199051
Adjacent sequences: A009113 A009114 A009115 * A009117 A009118 A009119
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KEYWORD
| sign,easy,nice
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AUTHOR
| R. H. Hardin (rhhardin(AT)att.net)
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EXTENSIONS
| Extended with signs Mar 15 1997 by Olivier Gerard.
Definition corrected, Joerg Arndt, Apr 29 2011.
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