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%I #63 Sep 08 2018 22:15:50
%S 1,1,4,20,128,1024,9856,110720,1421312,20525056,329334784,5812797440,
%T 111923560448,2334639652864,52444850814976,1262260748288000,
%U 32405895451246592,883950436237705216,25530268718794276864
%N Number of minimax trees on n nodes.
%C A minimax tree is (i) rooted, (ii) binary (i.e., each node has at most two sons), (iii) topological (i.e., the left son is different from the right son), (iv) labeled (i.e., there is a bijection between the nodes and a finite totally ordered set). Moreover it has the following property: (v) the label of each node x is the minimum or the maximum of all the labels of the nodes of the subtree whose root is x.
%H Vincenzo Librandi, <a href="/A080795/b080795.txt">Table of n, a(n) for n = 0..200</a>
%H D. Dominici, <a href="http://arxiv.org/abs/math/0501052">Nested derivatives: A simple method for computing series expansions of inverse functions.</a> arXiv:math/0501052v2 [math.CA]
%H Dominique Foata & Guo-Niu Han, <a href="http://www-irma.u-strasbg.fr/~foata/paper/pub86minimax.html">Arbres minimax et polynomes d'André </a>, Advances in Appl. Math., 27, 2001, p. 367-389.
%H Dominique Foata and Guo-Niu Han, <a href="http://dx.doi.org/10.1006/aama.2001.0740">Arbres minimax et polynomes d'André</a>. Special issue in honor of Dominique Foata's 65th birthday (Philadelphia, PA, 2000). Adv. in Appl. Math. 27 (2001), no. 2-3, 367-389.
%H E. Norton, <a href="http://arxiv.org/abs/1302.5411">Symplectic Reflection Algebras in Positive Characteristic as Ore Extensions</a>, arXiv preprint arXiv:1302.5411, 2013
%F E.g.f.: ( tanh(arctanh(sqrt(2)) - sqrt(2)*x) )/sqrt(2) = sqrt(2)/2* (1 + (3-2*sqrt(2))* exp(2*sqrt(2)*x) )/( 1 - (3-2*sqrt(2))* exp(2*sqrt(2)*x) ).
%F Recurrence: a(n+1) = 2*(Sum_{k=0..n} binomial(n,k)*a(k)*a(n-k)) - 0^n.
%F a(2*n) = 2^n * A006154(2*n), n>0 (conjectured). - _Ralf Stephan_, Apr 29 2004
%F For n>0, a(n) = sqrt(2)^(3*n+1)*Sum_{k>=0} k^n/(1+sqrt(2))^(2*k). - _Benoit Cloitre_, Jan 12 2005
%F From _Peter Bala_, Jan 30 2011: (Start)
%F A finite sum equivalent to the previous formula of _Benoit Cloitre_ is
%F a(n) = (2*sqrt(2))^(n-1)*Sum_{k = 1..n} k!*Stirling2(n,k)*w^(k-1), with w = (sqrt(2)-1)/2.
%F This formula can be used to prove congruences for a(n). For example,
%F a(p) = (-1)^((p^2-1)/8) (mod p) for odd prime p.
%F For similar formulas for labeled plane and non-plane unary-binary trees see A080635 and A000111 respectively.
%F For a sequence of related polynomials see A185419. For a recursive table to calculate a(n) see A185420.
%F The e.g.f. A(x) satisfies the autonomous differential equation
%F d/dx A(x) = 2*A(x)^2 - 1.
%F (End)
%F From _Peter Bala_, Aug 26 2011: (Start)
%F The inverse function A(x)^-1 of the generating function A(x) satisfies A(x)^-1 = Integral_{t = 0..x} 1/(2*t^2-1) dt. Let f(x) = 2*x^2-1. Define the nested derivative D^n[f](x) by means of the recursion D^0[f](x) = 1 and D^(n+1)[f](x) = d/dx(f(x)*D^n[f](x)) for n >= 0 (see A145271 for the coefficients in the expansion of D^n[f](x) in powers of f(x)). Then by [Dominici, Theorem 4.1] we have a(n+1) = D^n[f](1). For n >= 1 we have a(n) = (2+sqrt(2))^(n-1)*A(n,3-2*sqrt(2)), where {A(n,x)} n >= 1 = [1,1+x,1+4*x+x^2,1+11*x+11*x^2+x^3,...] denotes the sequence of Eulerian polynomials (see A008292). a(n+1) = (-1)^n*(sqrt(-2))^n*R(n,sqrt(-2)) where R(n,x) are the polynomials defined in A185896 (derivative polynomials associated with the function sec^2(x)). (End)
%F G.f.: 1 + x/G(0) where G(k) = 1 - 4*x*(k+1) - 2*x^2*(k+1)*(k+2)/G(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Jan 11 2013
%F G.f.: 1 + x/(G(0) -x), where G(k) = 1 - x*(k+1) - 2*x*(k+1)/(1 - x*(k+2)/G(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Dec 24 2013
%F E.g.f.: sqrt(2)*( -1/2 + (3+2*sqrt(2))/(4 + 2*sqrt(2)- E(0) )), where E(k) = 2 + 2*sqrt(2)*x/( 2*k+1 - 2*sqrt(2)*x/E(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Dec 27 2013
%F a(n) ~ n! * 2^((3*n+1)/2) / (log(3+2*sqrt(2)))^(n+1). - _Vaclav Kotesovec_, Feb 25 2014
%t Range[0, 18]! CoefficientList[ Series[ Tanh[ ArcTanh[ Sqrt[2]] - Sqrt[2] x]/Sqrt[2], {x, 0, 18}], x] (* _Robert G. Wilson v_ *)
%o (PARI) {Stirling2(n,k)=(1/k!)*sum(j=0,k,(-1)^j*binomial(k,j)*(k-j)^n)}
%o /* Finite sum given by _Peter Bala_: */
%o {a(n)=local(w=(sqrt(2)-1)/2);if(n==0,1,round((2*sqrt(2))^(n-1)*sum(k=1,n,k!*Stirling2(n,k)*w^(k-1))))}
%Y Cf. A185419, A185420. A008292, A185896.
%K nonn,easy
%O 0,3
%A _Emanuele Munarini_, Mar 14 2003
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