A006365 Number of binary tree partitions.

1, 1, 3, 8, 24, 72, 229, 740, 2460, 8317, 28590, 99556, 350661, 1246998, 4471801, 16153007, 58722226, 214687361, 788855637, 2911701588, 10790952975, 40139185202, 149805575022, 560804604385, 2105273566444, 7923577070408, 29892583708055, 113020768615090, 428190609376563, 1625319129822979, 6180270575440241

Offset

0,3

Links

Table of n, a(n) for n=0..30.

R. P. Stanley, A Fibonacci lattice, Fib. Quart., 13 (1975), 215-232.

Index entries for sequences related to rooted trees

Formula

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)).

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

Prog

(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 */
(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 */

Crossrefs

Sequence in context: A133787 A118264 A080923 * A178543 A188175 A046919

Adjacent sequences: A006362 A006363 A006364 * A006366 A006367 A006368

Keyword

nonn

Author

Don Knuth

Extensions

More terms from Joerg Arndt, May 18 2014

Status

approved