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

Author

Geoffrey Critzer, Apr 07 2012

Keywords

• nonn

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.

a(n) is the number of planar binary trees with leaves labeled by {1, ...,n} considered modulo swapping of the planar order at all the internal nodes except for the highest ones. - Bérénice Delcroix-Oger, Jun 25 2025

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

• Michael De Vlieger, Table of n, a(n) for n = 0..394
• Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
• Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 23.

Programs

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]

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, and (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

a(1)=1, a(2)=2 and a(n) = (1/2) * Sum_{k=1..n-1} C(n,k)*a(k)*a(n-k) for n>= 3. - Bérénice Delcroix-Oger, Jun 25 2025