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A006365
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Number of binary tree partitions.
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0
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1, 1, 3, 8, 24, 72, 229, 740, 2460, 8317, 28590, 99556, 350661, 1246998, 4471801, 16153007, 58722226, 214687361, 788855637, 2911701588, 10790952975
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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REFERENCES
| R. P. Stanley, A Fibonacci lattice, Fib. Quart., 13 (1975), 215-232.
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LINKS
| Index entries for sequences related to rooted trees
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FORMULA
| G.f.: G(z) = lim G_m(z), m -> inf, where G_m(z) = (2z^m)^(-1)*{1- sqrt( 1 - 4z^m Sum z^k*G_k(z)^2, 0<=k<m)}.
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
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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 */
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CROSSREFS
| Sequence in context: A133787 A080923 A118264 * A178543 A046919 A188175
Adjacent sequences: A006362 A006363 A006364 * A006366 A006367 A006368
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KEYWORD
| nonn
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AUTHOR
| D. E. Knuth.
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