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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

Index entries for sequences related to rooted trees

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 * A166882 A268937 A269634

Adjacent sequences:  A036771 A036772 A036773 * A036775 A036776 A036777

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

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 June 18 11:14 EDT 2018. Contains 305554 sequences. (Running on oeis4.)