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%I #54 Feb 18 2024 19:31:13
%S 1,-1,1,-3,13,-65,389,-2723,21785,-196065,1960649,-21567139,258805669,
%T -3364473697,47102631757,-706539476355,11304631621681,
%U -192178737568577,3459217276234385,-65725128248453315,1314502564969066301
%N Expansion of e.g.f. cos(x)/(1+x).
%C The absolute value of a(n) equals the real part of the permanent of the n X n matrix with (1+i)'s along the main diagonal, and 1's everywhere else. - _John M. Campbell_, Jul 10 2011
%H Robert Israel, <a href="/A009102/b009102.txt">Table of n, a(n) for n = 0..438</a>
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/IncompleteGammaFunction.html">Incomplete Gamma Function</a>.
%F a(n) = (-1)^n*round(n!*cos(1)). - _Vladeta Jovovic_, Aug 11 2002
%F a(n) = (-1)^n * n! * Sum_{k=0..floor(n/2} (-1)^k/(2k)!. Unsigned sequence satisfies e.g.f. cos(x)/(1-x). - _Ralf Stephan_, Apr 16 2004
%F E.g.f.: cos(x)/(1+x) = U(0)/(1-x^2) where U(k)= 1 - x/(1 - x/(x + (2*k+1)*(2*k+2)/U(k+1)) ; (continued fraction, 3-step). - _Sergei N. Gladkovskii_, Oct 17 2012
%F From _Vladimir Reshetnikov_, Oct 27 2015: (Start)
%F a(n) = Re((-i)^n*hypergeom([1,-n], [], i)).
%F a(n) = (-1)^n*(cos(1)*(n+2)!+cos(Pi*n/2)*hypergeom([1], [n/2+2,(n+3)/2], -1/4)+sin(Pi*n/2)*(n+2)*hypergeom([1], [n/2+1,(n+3)/2], -1/4))/(n^2+3*n+2).
%F a(n) = (-1)^n*Re(Gamma(n+1, i)*exp(i)) = (-1)^n*(Gamma(n+1, i)*exp(i)+Gamma(n+1, -i)*exp(-i))/2, where Gamma(a, x) is the upper incomplete Gamma function, i=sqrt(-1).
%F Gamma(n+1, i) = exp(-i)*((-1)^n*a(n) + A009551(n)*i).
%F Recurrence: a(0) = 1, a(1) = -1, a(2) = 1, a(n+3) = -(n+3)*a(n+2)-a(n+1)-(n+1)*a(n). (End)
%p G(x):=cos(x)/(1+x): f[0]:=G(x): for n from 1 to 20 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..20); # _Zerinvary Lajos_, Apr 03 2009
%p g:= gfun:-rectoproc({a(0) = 1, a(1) = -1, a(2) = 1, a(n+3) = -(n+3)*a(n+2)-a(n+1)-(n+1)*a(n)},a(n),remember):
%p seq(g(n),n=0..30); # _Robert Israel_, Oct 27 2015
%t Table[SeriesCoefficient[Cos[x]/(1+x), {x, 0, n}] n!, {n, 0, 20}]
%t Round@Table[(-1)^n Re[Gamma[n+1, I] E^I], {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 27 2015 *)
%t With[{nn=20},CoefficientList[Series[Cos[x]/(1+x),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Feb 18 2024 *)
%o (PARI) x='x+O('x^30); Vec(serlaplace(cos(x)/(1+x))) \\ _G. C. Greubel_, Jul 26 2018
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Cos(x)/(1+x))); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, Jul 26 2018
%Y Cf. A009551, A000142, A000166, A000522, A000023, A053486, A010844 (incomplete Gamma function values at other points).
%K sign,easy
%O 0,4
%A _R. H. Hardin_
%E Extended with signs by _Olivier Gérard_, Mar 15 1997
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