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%I #22 Jul 27 2018 08:56:47
%S 1,3,-7,9,177,-3897,65649,-1057851,16606977,-238404789,2305262889,
%T 33442089057,-3560906733903,182521828278351,-8055082800686367,
%U 338022326927690397,-13915405899740874879,566988435851123595411,-22784764731442383689127,888283409438427072329529
%N The n-th derivative of e^((2-x-x^2)/(1-x-x^2)), evaluated at x=1.
%C The n-th derivative of exp((2-x-x^2)/(1-x-x^2) is A(n,x) = n!*sum(m=1..n, sum(k=m..n, binomial(k-1,m-1)*binomial(k,n-k)*(2*x+1)^(2*k-n) * (-x^2-x+1)^(-m-k))/m!).
%H Alois P. Heinz, <a href="/A189244/b189244.txt">Table of n, a(n) for n = 0..388</a>
%H Vladimir Kruchinin, <a href="http://arxiv.org/abs/1104.5065">Derivation of Bell Polynomials of the Second Kind</a>, arXiv:1104.5065 [math.CO], 2011.
%F a(n) = n!*sum(m=1..n, sum(k=m..n, binomial(k-1,m-1) *binomial(k,n-k) * (-1)^(m+k)*3^(2*k-n))/m!), a(0)=1.
%F E.g.f.: exp(x*(3+x)/(3*x+x^2+1)). - _Alois P. Heinz_, Sep 27 2016
%t f[x_] := E^((2 - x - x^2)/(1 - x - x^2));
%t a[n_] := Derivative[n][f][1];
%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Jul 27 2018 *)
%o (Maxima)
%o a(n):=n!*sum(sum(binomial(k-1,m-1)*binomial(k,n-k)*(-1)^(m+k) * 3^(2*k-n), k,m,n)/m!,m,1,n)
%K sign
%O 0,2
%A _Vladimir Kruchinin_, Apr 26 2011
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