%I #41 Jun 28 2021 16:30:04
%S 1,1,2,4,10,26,73,211,630,1918,5944,18668,59311,190243,615269,2004025,
%T 6568174,21645438,71681152,238416580,796107464,2667768904,8968626418,
%U 30240087086,102238147891,346514952331,1177137322768,4007326361986,13669068510355,46711170248183,159899495303170
%N Revert transform of ((x - 1)(x + 1))/(-1 - x + x^3).
%C Binomial transform of aerated version of A001764. - _Paul Barry_, Nov 05 2006
%C a(n) is the number of planar rooted trees with n vertices, where each vertex has one or an even number of children. - Kurusch Ebrahimi-Fard and Dominique Manchon (kef(AT)unizar.es), Jun 05 2010
%H Michael De Vlieger, <a href="/A049130/b049130.txt">Table of n, a(n) for n = 1..1808</a>
%H Paul Barry, <a href="https://arxiv.org/abs/2104.01644">Centered polygon numbers, heptagons and nonagons, and the Robbins numbers</a>, arXiv:2104.01644 [math.CO], 2021.
%H K. Ebrahimi-Fard and D. Manchon, <a href="http://dx.doi.org/10.1016/j.jalgebra.2009.06.002">Dendriform Equations</a>, Journal of Algebra, 322 (2009), 4053-4079.
%H Aoife Hennessy, <a href="http://repository.wit.ie/1693/1/AoifeThesis.pdf">A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths</a>, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
%H <a href="/index/Res#revert">Index entries for reversions of series</a>
%F a(n) = sum{k=0..floor(n/2), C(n,2k)*C(3k,k)/(2k+1)}; a(n)=sum{k=0..n, C(3k/2,k/2)(1+(-1)^k)/(2(k+1))}. - _Paul Barry_, Nov 05 2006
%F G.f.: A(x) satisfies (x-1)*A(x)^3+(1-x)*A(x)-x = 0. - _Mark van Hoeij_, Apr 16 2013
%F Representation in terms of special values of hypergeometric function of type 4F3, in Maple notation a(n) = hypergeom([-n/2, (1-n)/2, 1/3, 2/3], [1, 1/2, 3/2], 27/4), n=0,1,2... . - _Karol A. Penson_, Jun 20 2013
%F Recurrence (for offset 0): -23*(n-3)*(n-2)*a(n-4) + 19*(n-2)*(2*n-3)*a(n-3) - 3*(n-2)*n*a(n-2) - 8*n*(2*n-1)*a(n-1) + 4*n*(n+1)*a(n) = 0. - _Vaclav Kotesovec_, Jun 28 2013
%F Asymptotics (for offset 0): a(n) ~ sqrt(3/Pi)*sqrt(1/2+1/(3*sqrt(3)))^3 * (1+3/2*sqrt(3))^n/n^(3/2). - _Vaclav Kotesovec_, Jun 28 2013
%F G.f.: 1/(1-x - 3*x^2/(S(0)*(1-x))), where S(k) = 4*k+3 - 3*x^2*(3*k+4)*(6*k+5)/( 2*(1-x)^2*(4*k+5) - 3*x^2*(3*k+5)*(6*k+7)/S(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Dec 26 2013
%p series(RootOf((x-1)*A^3+(1-x)*A-x,A),x=0,30); # _Mark van Hoeij_, Apr 16 2013
%t Table[Sum[Binomial[n,2*k]*Binomial[3*k,k]/(2*k+1),{k,0,Floor[n/2]}],{n,0,20}] (* _Vaclav Kotesovec_, Jun 28 2013 *)
%o (PARI) Vec( serreverse(x*(1-x)*(1+x)/(1+x-x^3) +O(x^66) ) ) \\ _Joerg Arndt_, Sep 12 2013
%K nonn
%O 1,3
%A _Olivier GĂ©rard_