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%I #68 Aug 02 2019 17:01:14
%S 1,2,22,584,28384,2190128,245762848,37788392576,7625538720256,
%T 1954588198280192,620259836756837632,238698984906300222464,
%U 109521341941344601083904,59065100769855968517951488,36990397033719114096675954688
%N E.g.f. A(x) satisfies A(x) = (1 + (Integral A(x) dx)^2 / 2)^2.
%C Previous name was: A mysterious sequence.
%C This is based on the derivatives of the real function g(x) := -1/f(x)^2:
%C The algorithm for the sequence is as follows.
%C (1) Dj = 0, for each j, when j is odd (j=2k+1); (odd derivatives are null)
%C (3) D2 = -1*f(a)^-2; then b1 = 1; (the 2nd derivative)
%C (4) D4 = -2*f(a)^-5; (the 4th derivative) So b2 = 2;
%C (5) D6 = -22*f(a)^-8; (the 6th derivative) So b3 = 22;
%C (6) D8 = -584*f(a)^-11 (the 8th derivative) So b4 = 584;
%C (8) D10= -28384*f(a)^-14 (the 10th derivative) So b5 = 28384; and so on...
%C (n) D2n= -bn*f(a)^-(3n-1) (the 2n-th derivative) on general bn is unknown.
%C a(n) = [x^(2n) / (2n)!] A(x). A(-x) = A(x). - _Michael Somos_, Aug 26 2014
%C Number of 2-bundled bilabeled increasing trees with 2n labels. - _Markus Kuba_, Nov 18 2014
%H Alois P. Heinz and Vaclav Kotesovec, <a href="/A120419/b120419.txt">Table of n, a(n) for n = 0..226</a> (first 100 terms from Alois P. Heinz)
%H Markus Kuba and Alois Panholzer, <a href="http://arxiv.org/abs/1411.4587">Combinatorial families of multilabelled increasing trees and hook-length formulas</a>, arXiv:1411.4587 [math.CO], 2014.
%H Markus Kuba and Alois Panholzer, <a href="https://doi.org/10.1016/j.disc.2015.08.010">Combinatorial families of multilabelled increasing trees and hook-length formulas</a>, Discrete Mathematics 339(1) (2016), 227-254.
%H H. Sussmann, <a href="http://math.rutgers.edu/~sussmann/papers/smf-paris-June-2000.pdf">Résultats récents sur les courbes optimales</a>, Soc. Math. de France du 17 juin 2000.
%H H. Sussmann and J. C. Willems, <a href="https://doi.org/10.1109/37.588098">300 Years of Optimal Control: from the brachystochrone to the maximum principle</a>, IEEE Control Systems, 17(3) (1997), 32-44.
%H H. Sussmann and J. C. Willems, <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.26.825&rep=rep1&type=pdf">The Brachistochrone Problem and Modern Control Theory</a>, University of Groningen, May 1999.
%H H. Sussmann and J. C. Willems, <a href="https://doi.org/10.1142/9789812778079_0004">The Brachistochrone Problem and Modern Control Theory</a>, in: Contemporary Trends in Nonlinear Geometric Control Theory and Its Applications (A. Anzaldo-Meneses, F. Monroy-Pérez, B. Bonnard, and J. P. Gauthier, eds.), pp. 113-166, 2002.
%F E.g.f. A(x) satisfies: A(x) = (1 + Integral (A(x) * Integral A(x) dx) dx)^2. - _Paul D. Hanna_, Aug 26 2014
%F E.g.f. A(x) satisfies: A'(x) = 2*A(x)^(3/2) * Integral A(x) dx. - _Paul D. Hanna_, Aug 26 2014
%F Note that the e.g.f. for Euler numbers (A000364) satisfies G(x) = 1 + Integral (G(x) * Integral G(x)^2 dx) dx when G(x) = 1/cos(x). - _Paul D. Hanna_, Aug 26 2014
%F E.g.f.: (1 + Series_Reversion( sqrt(2)*( atan(x) + x/(1+x^2) )/2 )^2 )^2. - _Paul D. Hanna_, Aug 26 2014, after rewriting a formula due to Robert Israel.
%F E.g.f. A(x) satisfies A(x) = (1 + (Integral A(x) dx)^2 / 2)^2. - _Michael Somos_, Aug 26 2014
%F Limit n->infinity (a(n)/(2*n)!)^(1/n) = 8/Pi^2. - _Vaclav Kotesovec_, Nov 18 2014
%F E.g.f. (for offset 1) T=T(z) satisfies T''=1/(1-T)^2; an implicit equation for T is 2*(arcsin(sqrt(T))+sqrt(T(1-T)))=z^2. - _Markus Kuba_ Nov 18 2014
%e E.g.f.: A(x) = 1 + 2*x^2/2! + 22*x^4/4! + 584*x^6/6! + 28384*x^8/8! +...
%t a[ n_] := If[ n < 1, Boole[n == 0], With[{m = 2 n - 1}, m! SeriesCoefficient[ InverseSeries[ Integrate[ Series[ (1 + x^2/2)^-2, {x, 0, m}], x]], {x, 0, m}]]]; (* _Michael Somos_, Aug 26 2014 *)
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=(1+intformal(A*intformal(A +x*O(x^n))))^2 );n!*polcoeff(A,n)}
%o for(n=0,20,print1(a(2*n),", ")) \\ _Paul D. Hanna_, Aug 26 2014
%o (PARI) {a(n)=local(A);A=(1 + serreverse( sum(m=1,n\2+1,(-1/2)^(m-1) * m * x^(2*m-1) / (2*m-1)) +x^2*O(x^n) )^2/2)^2;n!*polcoeff(A,n)}
%o for(n=0,20,print1(a(2*n),", ")) \\ _Paul D. Hanna_, Aug 26 2014
%o (PARI) {a(n) = if( n<1, n==0, n*=2; (n-1)! * polcoeff( serreverse( intformal( (1 + x^2 / 2 + O(x^n))^-2)), n-1))}; /* _Michael Somos_, Aug 26 2014 */
%K nonn
%O 0,2
%A Robert Wackensack (wackensack(AT)hotmail.com), Jul 09 2006
%E New name from _Paul D. Hanna_, Aug 26 2014
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