This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A182037 #54 Jun 26 2025 01:20:28
%S A182037 0,1,2,6,36,300,3240,42840,670320,12111120,248119200,5683154400,
%T A182037 143910043200,3991909521600,120376874217600,3920816403504000,
%U A182037 137177166174048000,5130755025780384000,204295093225134912000,8627985710304472512000,385222786392984059520000
%N A182037 Expansion of 1 - (1 - 2*x - x^2)^(1/2).
%C A182037 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.
%C A182037 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
%C A182037 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
%H A182037 Michael De Vlieger, <a href="/A182037/b182037.txt">Table of n, a(n) for n = 0..394</a>
%H A182037 Alexander Burstein and Louis W. Shapiro, <a href="https://arxiv.org/abs/2112.11595">Pseudo-involutions in the Riordan group</a>, arXiv:2112.11595 [math.CO], 2021.
%H A182037 Bérénice Delcroix-Oger and Clément Dupont, <a href="https://arxiv.org/abs/2505.06094">Lie-operads and operadic modules from poset cohomology</a>, arXiv:2505.06094 [math.CO], 2025. See p. 23.
%F A182037 E.g.f.: 1 - (1-2*x-x^2)^(1/2).
%F A182037 E.g.f.: x*(1+A(x)) where A(x) is the e.g.f. of A036774.
%F A182037 a(n) ~ sqrt(2-sqrt(2)) * n^(n-1) * (1+sqrt(2))^n / exp(n). - _Vaclav Kotesovec_, Sep 25 2013
%F A182037 From _Benedict W. J. Irwin_, May 25 2016: (Start)
%F A182037 Let y(0)=1, y(1)=-1, and (1-n)*y(n) - (2n+1)*y(n+1) + (n+2)*y(n+2) = 0,
%F A182037 a(n) = -n!y(n), n > 0. (End)
%F A182037 a(n) + (-2*n+3)*a(n-1) - (n-1)*(n-3)*a(n-2) = 0. - _R. J. Mathar_, Jun 08 2016
%F A182037 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
%t A182037 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]
%K A182037 nonn
%O A182037 0,3
%A A182037 _Geoffrey Critzer_, Apr 07 2012