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A005264
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Number of labeled rooted Greg trees with n nodes.
(Formerly M3096)
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9
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1, 3, 22, 262, 4336, 91984, 2381408, 72800928, 2566606784, 102515201984, 4575271116032, 225649908491264, 12187240730230528, 715392567595403520, 45349581052869924352, 3087516727770990992896, 224691760916830871873536
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| A rooted Greg tree can be described as a rooted tree with 2-colored nodes where only the black nodes are counted and labeled and the white nodes have at least 2 children. (Christian G. Bower, Nov 15 1999)
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REFERENCES
| J. Felsenstein, The number of evolutionary trees, Systematic Zoology, 27 (1978), 27-33.
C. Flight, How many stemmata?, Manuscripta, 34 (1990), 122-128.
L. R. Foulds and R. W. Robinson, Determining the asymptotic number of phylogenetic trees, pp. 110-126 of Combinatorial Mathematics VII (Newcastle, August 1979), ed. R. W. Robinson, G. W. Southern and W. D. Wallis. Lect. Notes in Math., 829. Springer, 1980.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| Index entries for reversions of series
Index entries for sequences related to rooted trees
Index entries for sequences related to trees
D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions. arXiv:math/0501052v2 [math.CA]
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FORMULA
| Exponential reversion of A157142 with offset 1. - Michael Somos Mar 26 2011
E.g.f. A(x) = y satisfies y' = (1 + 2*y) / ((1 - 2*y) * (1 + x)). - Michael Somos Mar 26 2011
E.g.f. A(x) satisfies (1 + x) * exp(A(x)) = 1 + 2 * A(x).
From Peter Bala, Sep 08 2011: (Start)
A(x) satisfies the separable differential equation A'(x) = exp(A(x))/(1-2*A(x)) with A(0) = 0. Thus the inverse function A^-1(x) = int {t = 0..x} (1-2*t)/exp(t) = exp(-x)*(2*x+1)-1 = x-3*x^2/2!+5*x^3/3!-7*x^4/4!+.... A(x) = -1/2-LambertW(-exp(-1/2)*(x+1)/2).
The expansion of A(x) can be found by inverting the above integral using the method of [Dominici, Theorem 4.1] to arrive at the result a(n) = D^(n-1)(1) evaluated at x = 0, where D denotes the operator g(x) -> d/dx(exp(x)/(1-2*x)*g(x)). Compare with [Dominici, Example 9].
(End)
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EXAMPLE
| x + 3*x^2 + 22*x^3 + 262*x^4 + 4336*x^5 + 91984*x^6 + 2381408*x^7 + ...
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PROG
| (PARI) {a(n) = local(A); if( n<1, 0, for( k= 1, n, A += x * O(x^k); A = truncate( (1 + x) * exp(A) - 1 - A) ); n!* polcoeff( A, n))} /* Michael Somos Apr 02 2007 */
(PARI) {a(n) = if( n<1, 0, n! * polcoeff( serreverse( exp( -x + x * O(x^n) ) * (1 + 2*x) - 1), n))} /* Michael Somos Mar 26 2011 */
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CROSSREFS
| Inverse Stirling transform of A005172 (hence corrected and extended) - John Layman (layman(AT)calvin.math.vt.edu).
Cf. A005263, A048159, A048160, A052300-A052303, A157142.
Sequence in context: A143634 A054595 A054594 * A195512 A052892 A155806
Adjacent sequences: A005261 A005262 A005263 * A005265 A005266 A005267
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KEYWORD
| nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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