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%I #25 Jul 21 2018 23:38:16
%S 1,0,-1,3,-6,0,90,-630,2520,0,-113400,1247400,-7484400,0,681080400,
%T -10216206000,81729648000,0,-12504636144000,237588086736000,
%U -2375880867360000,0,548828480360160000,-12623055048283680000
%N Expansion of E.g.f.: (1 + x)/(1 + x + x^2/2).
%H G. C. Greubel, <a href="/A009014/b009014.txt">Table of n, a(n) for n = 0..450</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.
%F E.g.f.: (1+x)/(1+x+x^2/2) = 1/cosh(log(1+x)).
%F a(n) = -n*a(n-1)-n*(n-1)/2*a(n-2), a(0)=1, a(1)=0.
%F From _Peter Bala_, Oct 05 2011: (Start)
%F a(n) = -n!*(((-1+i)/2)^(n+1) + ((-1-i)/2)^(n+1)) = -n!/sqrt(2)^(n-1)*cos(3*Pi*(n+1)/4).
%F a(n) = 1/2^n*A009769(n) = -n!/2^n*A009116(n+1).
%F a(n) = 2*(-1)^n*int {t = 0..inf} t^n*exp(-t)*cos(t). (End)
%p seq(coeff(series(factorial(n)*((1+x)/(1+x+x^2/2)), x,n+1),x,n),n=0..25); # _Muniru A Asiru_, Jul 21 2018
%t nn = 26; Range[0, nn]! CoefficientList[Series[1/Cosh[Log[1 + x]], {x, 0, nn}], x] (* _T. D. Noe_, Oct 05 2011 *)
%o (PARI) a(n)=-(2^-n)*n!*real((-1+I)^(n+1))
%o (PARI) x='x+O('x^30); Vec(serlaplace(1/cosh(log(1+x)))) \\ _G. C. Greubel_, Jul 21 2018
%Y Cf. A009116, A009769.
%K sign,easy
%O 0,4
%A _R. H. Hardin_
%E Extended with signs by _Olivier Gérard_, Mar 15 1997
%E Better description from _Michael Somos_
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