OFFSET
0,2
COMMENTS
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).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..388
Vladimir Kruchinin, Derivation of Bell Polynomials of the Second Kind, arXiv:1104.5065 [math.CO], 2011.
Antoine Genitrini, Bernhard Gittenberger, Manuel Kauers and Michael Wallner, Asymptotic enumeration of compacted binary trees of bounded right height, arXiv:1703.10031 [math.CO], 2017; J. Combin. Theory Ser. A 172 (2020), 105177, 49 pp.
FORMULA
a(n) = n!*Sum_{k=1..n} (-1)^(k+1)*binomial(k,n-k)*3^(2*k-n), a(0)=-1.
E.g.f.: -1/(x^2+3*x+1). - Alois P. Heinz, Sep 27 2016
D-finite with recurrence: n*(n-1)*a(n-2) + 3*n*a(n-1) + a(n) = 0. - Georg Fischer, Aug 18 2021
a(n) = (-1)^(n+1)*n!*A001906(n+1); see Theorem 7.8 in [Genitrini et al, 2020] - Michael Wallner, Jul 13 2023.
MAPLE
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
MATHEMATICA
f[x_] := 1/(1 - x - x^2);
a[n_] := Derivative[n][f][1];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jul 27 2018 *)
PROG
(Maxima)
a(n):=n!*sum((-1)^(k+1)*binomial(k, n-k)*3^(2*k-n), k, 1, n);
CROSSREFS
KEYWORD
sign
AUTHOR
Vladimir Kruchinin, Apr 26 2011
STATUS
approved