A306557 Numerator coefficients of the bivariate Maclaurin series ("inverse Kepler equation") developed as Lagrange inversion E=KeplerInv(e,M) of Kepler's equation M = Kepler(e,E) = E - e*sin(E).

1, 1, 9, 1, 54, 225, 1, 243, 4131, 11025, 1, 1008, 50166, 457200, 893025, 1, 4077, 520218, 11708154, 70301925, 108056025, 1, 16362, 5020623, 243313164, 3274844175, 14427513450, 18261468225, 1, 65511, 46789461, 4535570691, 119537963811, 1107456067125, 3821273720775, 4108830350625

Offset

0,3

Comments

Coefficients of the numerator polynomials of the bivariate Maclaurin series ("inverse Kepler equation") developed as Lagrange inversion E = KeplerInv(e,M) of Kepler's equation M = Kepler(e,E) = E - e*sin(E), where e=numeric eccentricity, M=mean anomaly, E=eccentric anomaly. The series is KeplerInv(e,M) = M/(1-e) + Sum_{n>=1} (-1)^n*(Sum_{j=1..n} a(n,j)*e^j)/(1-e)^(3n+1)*M^(2n+1)/(2n+1)! = M/(1-e) - (e/(1-e)^4)*M^3/3! + ((e+9*e^2)/(1-e)^7)*M^5/5! - + ... .

The element a(n,n) with highest index in each row (the diagonal element) has the form Product_{j=1..n} (2*j+1)^2.

The derivative dKepler/dE = 1 - e*cos(E) goes to zero at E = i*arccosh(1/e) in the complex plane. Thus dKeplerInv/dM goes to infinity at M = i*(arccosh(1/e) - sqrt(1-e^2)), so that the radius of convergence of KeplerInv(e,M) is arccosh(1/e) - sqrt(1-e^2). KeplerInv(e,M) converges linearly within the circle of convergence |M| < arccosh(1/e) - sqrt(1-e^2).

Links

Table of n, a(n) for n=0..35.

Wikipedia, Kepler's equation

Formula

While M = E - e*sin(E) = E*(1-e) - e*Sum_{n>=1} (-1)^n*E^(2n+1)/(2n+1)! the formal power series of the compositional inverse KeplerInv(e,M) is as above according to A111785 and A304462.

Example

Matrix (regular triangle) lexicographically ascending in the rows:
  1;
  1,     9;
  1,    54,     225;
  1,   243,    4131,     11025;
  1,  1008,   50166,    457200,     893025;
  1,  4077,  520218,  11708154,   70301925,   108056025;
  1, 16362, 5020623, 243313164, 3274844175, 14427513450, 18261468225;
  ...

Crossrefs

Generated by A111785 or A304462, diagonal elements are in A001818.

Sequence in context: A283016 A050303 A286920 * A283060 A283082 A318935

Adjacent sequences: A306554 A306555 A306556 * A306558 A306559 A306560

Keyword

nonn,tabl

Author

Herbert Eberle, Feb 23 2019

Status

approved