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 A036774 Number of labeled rooted unordered binary trees (each node has out-degree <=2). 9
 0, 1, 2, 9, 60, 540, 6120, 83790, 1345680, 24811920, 516650400, 11992503600, 307069963200, 8598348158400, 261387760233600, 8573572885878000, 301809119163552000, 11349727401396384000, 454104511068656448000, 19261139319649202976000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 REFERENCES L. Takacs, Enumeration of rooted trees and forests, Math. Scientist 18 (1993), 1-10, esp. Eq. (14) with r = 2. LINKS Fung Lam, Table of n, a(n) for n = 0..394 FORMULA E.g.f. (1-x-sqrt(1-2*x-x^2))/x. E.g.f. A(x) satisfies x*A(x)^2 +2*(x-1)*A(x)+2*x=0, A(0)=0 and A(x)=x/(1-x-(x/2)*A(x)). - Michael Somos, Sep 06 2003 a(n) = n!*sum(binomial(n-1, 2k)*binomial(2k, k)/(2^k*(k+1)), k=0..floor((n-1)/2)). [Emanuele Munarini, Feb 06 2013] a(n) ~ sqrt(2-sqrt(2))*n^(n-1)/(exp(n)*(sqrt(2)-1)^(n+1)). - Vaclav Kotesovec, Sep 24 2013 Recurrence: (n+1)*a(n) = n*(n-1)*(n-2)*a(n-2) + n*(2*n-1)*a(n-1), n>=3, a(1)=1, a(2)=2. - Fung Lam, Feb 24 2014 MATHEMATICA Range[0, 20]! CoefficientList[Series[(1 - x - ((x - 1)^2 - 2 x^2)^(1/2))/x, {x, 0, 20}], x] (*Geoffrey Critzer, Nov 22 2011*) PROG (PARI) a(n)=if(n<1, 0, n!*polcoeff(2*x/(1-x+sqrt(1-2*x-x^2+O(x^n))), n)) (PARI) a(n)=if(n<1, 0, n!*polcoeff(serreverse(2*x/(2+2*x+x^2)+x*O(x^n)), n)) (Maxima) makelist(n!*sum(binomial(n-1, 2*k)*binomial(2*k, k)/(2^k*(k+1)), k, 0, floor((n-1)/2)), n, 0, 20); [Emanuele Munarini, Feb 06 2013] CROSSREFS A071356(n)=a(n+1)2^n/(n+1)!. Sequence in context: A001193 A161391 A120014 * A306065 A166882 A268937 Adjacent sequences:  A036771 A036772 A036773 * A036775 A036776 A036777 KEYWORD nonn AUTHOR EXTENSIONS Better description and formula from Christian G. Bower, Nov 29 2001 Added "unordered" to the name, David Callan, Apr 22 2012 STATUS approved

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Last modified August 20 08:50 EDT 2018. Contains 313914 sequences. (Running on oeis4.)