A188805 The n-th derivative of 1/(1-x-x^2), evaluated at x=1.

-1, 3, -16, 126, -1320, 17280, -271440, 4974480, -104186880, 2454883200, -64269676800, 1850862182400, -58147441228800, 1979015707468800, -72535825410048000, 2848518844883712000, -119320306456006656000, 5310538503447969792000

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

Cf. A001906.

Sequence in context: A246525 A193242 A247591 * A214645 A296535 A088358

Adjacent sequences: A188802 A188803 A188804 * A188806 A188807 A188808

Keyword

sign

Author

Vladimir Kruchinin, Apr 26 2011

Status

approved