A182037 Expansion of 1 - (1 - 2*x - x^2)^(1/2).

0, 1, 2, 6, 36, 300, 3240, 42840, 670320, 12111120, 248119200, 5683154400, 143910043200, 3991909521600, 120376874217600, 3920816403504000, 137177166174048000, 5130755025780384000, 204295093225134912000, 8627985710304472512000, 385222786392984059520000

Offset

0,3

Comments

a(n) is the number of rooted labeled trees such that (i) the root vertex has at most one child and (ii) all other vertices have at most two children.

F(x) = -e.g.f. (below) = -1 + (2-(1+x)^2)^(1/2) is self-inverse about x=0, i.e., its own compositional inverse, so the negative of the integer sequence remains unchanged by Lagrange inversion. This results from viewing y=F(x) as describing the arc, in the second and fourth quadrant, of a circle centered at (-1,-1) with radius sqrt(2). - Tom Copeland, Oct 05 2012

Links

Table of n, a(n) for n=0..20.

Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.

Formula

E.g.f.: 1 - (1-2*x-x^2)^(1/2).

E.g.f.: x*(1+A(x)) where A(x) is the e.g.f. of A036774.

a(n) ~ sqrt(2-sqrt(2)) * n^(n-1) * (1+sqrt(2))^n / exp(n). - Vaclav Kotesovec, Sep 25 2013

From Benedict W. J. Irwin, May 25 2016: (Start)

Let y(0)=1, y(1)=-1,

Let (1-n)y(n) - (2n+1)y(n+1) + (n+2)y(n+2) = 0,

a(n) = -n!y(n), n > 0.

(End)

a(n) + (-2*n+3)*a(n-1) - (n-1)*(n-3)*a(n-2) = 0. - R. J. Mathar, Jun 08 2016

Mathematica

nn = 15; a = (1 - x - (1 - 2 x - x^2)^(1/2))/x; Range[0, nn]! CoefficientList[Series[x + a x, {x, 0, nn}], x]

Crossrefs

Sequence in context: A234235 A277740 A277393 * A306066 A061302 A055541

Adjacent sequences: A182034 A182035 A182036 * A182038 A182039 A182040

Keyword

nonn

Author

Geoffrey Critzer, Apr 07 2012

Status

approved