|
%I #53 Aug 09 2023 12:54:48
%S -1,3,-16,126,-1320,17280,-271440,4974480,-104186880,2454883200,
%T -64269676800,1850862182400,-58147441228800,1979015707468800,
%U -72535825410048000,2848518844883712000,-119320306456006656000,5310538503447969792000
%N The n-th derivative of 1/(1-x-x^2), evaluated at x=1.
%C The n-th derivative of 1/(1-x-x^2) is A(n,x) = n!*Sum_{k=1..n} binomial(k,n-k)*(2*x+1)^(2*k-n)*(-x^2-x+1)^(-k-1).
%H Alois P. Heinz, <a href="/A188805/b188805.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.
%H Antoine Genitrini, Bernhard Gittenberger, Manuel Kauers and Michael Wallner, <a href="https://arxiv.org/abs/1703.10031">Asymptotic enumeration of compacted binary trees of bounded right height</a>, arXiv:1703.10031 [math.CO], 2017; J. Combin. Theory Ser. A 172 (2020), 105177, 49 pp.
%F a(n) = n!*Sum_{k=1..n} (-1)^(k+1)*binomial(k,n-k)*3^(2*k-n), a(0)=-1.
%F E.g.f.: -1/(x^2+3*x+1). - _Alois P. Heinz_, Sep 27 2016
%F D-finite with recurrence: n*(n-1)*a(n-2) + 3*n*a(n-1) + a(n) = 0. - _Georg Fischer_, Aug 18 2021
%F a(n) = (-1)^(n+1)*n!*A001906(n+1); see Theorem 7.8 in [Genitrini et al, 2020] - _Michael Wallner_, Jul 13 2023.
%p f:= gfun:-rectoproc({n*(n-1)*a(n-2)+3*n*a(n-1)+a(n), a(0)=-1, a(1)=3}, a(n), remember): map(f, [$0..20]); # _Georg Fischer_, Aug 18 2021
%t f[x_] := 1/(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((-1)^(k+1)*binomial(k,n-k)*3^(2*k-n),k,1,n);
%Y Cf. A001906.
%K sign
%O 0,2
%A _Vladimir Kruchinin_, Apr 26 2011
|
