A120419 E.g.f. A(x) satisfies A(x) = (1 + (Integral A(x) dx)^2 / 2)^2.

1, 2, 22, 584, 28384, 2190128, 245762848, 37788392576, 7625538720256, 1954588198280192, 620259836756837632, 238698984906300222464, 109521341941344601083904, 59065100769855968517951488, 36990397033719114096675954688

Offset

0,2

Comments

Previous name was: A mysterious sequence.

This is based on the derivatives of the real function g(x) := -1/f(x)^2:

The algorithm for the sequence is as follows.

(1) Dj = 0, for each j, when j is odd (j=2k+1); (odd derivatives are null)

(3) D2 = -1*f(a)^-2; then b1 = 1; (the 2nd derivative)

(4) D4 = -2*f(a)^-5; (the 4th derivative) So b2 = 2;

(5) D6 = -22*f(a)^-8; (the 6th derivative) So b3 = 22;

(6) D8 = -584*f(a)^-11 (the 8th derivative) So b4 = 584;

(8) D10= -28384*f(a)^-14 (the 10th derivative) So b5 = 28384; and so on...

(n) D2n= -bn*f(a)^-(3n-1) (the 2n-th derivative) on general bn is unknown.

a(n) = [x^(2n) / (2n)!] A(x). A(-x) = A(x). - Michael Somos, Aug 26 2014

Number of 2-bundled bilabeled increasing trees with 2n labels. - Markus Kuba, Nov 18 2014

Links

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..226 (first 100 terms from Alois P. Heinz)

Markus Kuba and Alois Panholzer, Combinatorial families of multilabelled increasing trees and hook-length formulas, arXiv:1411.4587 [math.CO], 2014.

Markus Kuba and Alois Panholzer, Combinatorial families of multilabelled increasing trees and hook-length formulas, Discrete Mathematics 339(1) (2016), 227-254.

H. Sussmann, Résultats récents sur les courbes optimales, Soc. Math. de France du 17 juin 2000.

H. Sussmann and J. C. Willems, 300 Years of Optimal Control: from the brachystochrone to the maximum principle, IEEE Control Systems, 17(3) (1997), 32-44.

H. Sussmann and J. C. Willems, The Brachistochrone Problem and Modern Control Theory, University of Groningen, May 1999.

H. Sussmann and J. C. Willems, The Brachistochrone Problem and Modern Control Theory, 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.

Formula

E.g.f. A(x) satisfies: A(x) = (1 + Integral (A(x) * Integral A(x) dx) dx)^2. - Paul D. Hanna, Aug 26 2014

E.g.f. A(x) satisfies: A'(x) = 2*A(x)^(3/2) * Integral A(x) dx. - Paul D. Hanna, Aug 26 2014

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

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.

E.g.f. A(x) satisfies A(x) = (1 + (Integral A(x) dx)^2 / 2)^2. - Michael Somos, Aug 26 2014

Limit n->infinity (a(n)/(2*n)!)^(1/n) = 8/Pi^2. - Vaclav Kotesovec, Nov 18 2014

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

Example

E.g.f.: A(x) = 1 + 2*x^2/2! + 22*x^4/4! + 584*x^6/6! + 28384*x^8/8! +...

Mathematica

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 *)

Prog

(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)}
for(n=0, 20, print1(a(2*n), ", ")) \\ Paul D. Hanna, Aug 26 2014
(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)}
for(n=0, 20, print1(a(2*n), ", ")) \\ Paul D. Hanna, Aug 26 2014
(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 */

Crossrefs

Sequence in context: A328020 A246740 A248798 * A217912 A210657 A177042

Adjacent sequences: A120416 A120417 A120418 * A120420 A120421 A120422

Keyword

nonn

Author

Robert Wackensack (wackensack(AT)hotmail.com), Jul 09 2006

Extensions

New name from Paul D. Hanna, Aug 26 2014

Status

approved