%I #24 Jan 15 2022 04:39:04
%S 1,1,3,8,24,72,229,740,2460,8317,28590,99556,350661,1246998,4471801,
%T 16153007,58722226,214687361,788855637,2911701588,10790952975,
%U 40139185202,149805575022,560804604385,2105273566444,7923577070408,29892583708055,113020768615090,428190609376563,1625319129822979,6180270575440241
%N Number of binary tree partitions.
%H R. P. Stanley, <a href="http://www.fq.math.ca/Scanned/13-3/stanley.pdf">A Fibonacci lattice</a>, Fib. Quart., 13 (1975), 215-232.
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F G.f.: G(z) = lim_{m->infinity} G_m(z), where G_m(z) = (2z^m)^(-1)*(1 - sqrt(1 - 4z^m * Sum_{k=0..m-1} z^k*G_k(z)^2)).
%F Given the AGM-like recursion f(a0,b0,c0) = (a1,b1,c1) where a0^2 = b0^2 + 2*a0*c0, a1^2 = b1^2 + 2*a1*c1, a1 = (a0 + b0)/2, c1=c0*x with initial values a0=1, c0=2*x, then the common limit of a and b is 1/A(x). - _Michael Somos_, Sep 18 2006
%o (PARI) {a(n)=local(A); if(n<0, 0, A=1+x*O(x^n); for(k=1, n, A=(1-sqrt(1-4*x*A))/2); polcoeff(A, 2*n))} /* _Michael Somos_, Sep 18 2006 */
%o (PARI) {a(n)=local(A); if(n<0, 0, A=1+x*O(x^n); for(k=1, n, A*=2/(1+sqrt(1-A*4*x^k))); polcoeff(A, n))} /* _Michael Somos_, Sep 18 2006 */
%K nonn
%O 0,3
%A _Don Knuth_
%E More terms from _Joerg Arndt_, May 18 2014