

A335828


Numerators of coefficients in a power series expansion of the distance between two bodies falling freely towards each other along a straight line under the influence of their mutual gravitational attraction.


2



1, 1, 11, 73, 887, 136883, 7680089, 26838347, 14893630313, 1908777537383, 2422889987331397, 233104477447558811, 2430782624763507659, 14420190617640617313953, 4515429325405165295004389, 812454316441781379614873497, 166481868581561511154267399013
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OFFSET

1,3


COMMENTS

Consider two point objects with masses m_1 and m_2 that are starting to fall towards each other from rest at time t = 0 and initial distance r_0. Foong (2008) gave the solution for the distance as a function of time, r(t) = r_0 * f(t/t_0), where t_0 = sqrt(r_0^3/(G*(m1+m2)), G is the gravitational constant (A070058), and f(x) = 1  Sum_{n>=1} c(n) * x^(2*n) is a dimensionless function. c(n) are the rational coefficients whose numerators are given in this sequence. The denominators are given in A335829. The collision occurs when f(x) = 0, at x = Pi/(2*sqrt(2)) (A093954), which corresponds to the time t = (Pi/(2*sqrt(2))) * t_0.
A similar expansion was given by Ernst Meissel in his study of the threebody problem in 1882. In Meissel's expansion the coefficients are c(n)/2^n.


REFERENCES

Sudhir Ranjan Jain, Mechanics, Waves and Thermodynamics: An Examplebased Approach, Cambridge University Press, 2016. See page 97.
Ernst Meissel, Über Reihen, denen man bei der numerischen Lösung des Problems der Dreikörperproblems begegnet, wenn die Anfangsgeschwindigkeiten Null sind, in: Jahresbericht über die Realschule in Kiel: Während des Schuljahres 1881/82, A. F. Jensen, Kiel, 1882, pp. 111.


LINKS

Amiram Eldar, Table of n, a(n) for n = 1..243
S. K. Foong, From Moonfall to motions under inverse square laws, European journal of physics, Vol. 29, No. 5 (2008), pp. 9871003, alternative link.
Jaak Peetre, Ernst Meissel and the Pythagorean problem  the DreiKörperProblem in the Nachlass Meissel, draft, 1997.


FORMULA

a(n) = numerator(c(n)), c(1) = 1/2, c(n) = (2 * Sum_{k=1..n1} (nk)*(2*n2*k1)*c(nk)*c(k)  Sum_{m=2..n1} (nm)*(2*n2*m1)*c(nm) * Sum_{k=1..m1} c(mk)*c(k))/(n*(2*n  1)).
c(n) ~ c_0 * n^(5/3) * (Pi/(2*sqrt(2))^(2*n), where c_0 = (3*Pi)^(2/3) / (18*Gamma(4/3)) = 0.277587...


EXAMPLE

The series begins with f(x) = 1  (1/2)*x^2  (1/12)*x^4  (11/360)*x^6  ...


MATHEMATICA

c[1] = 1/2; c[n_] := c[n] = (2*Sum[(n  k)*(2*n  2*k  1)*c[n  k]*c[k], {k, 1, n  1}]  Sum[(n  m)*(2*n  2*m  1)*c[n  m]*c[m  k]*c[k], {m, 2, n  1}, {k, 1, m  1}])/(n*(2*n  1)); Numerator @ Array[c, 17]
(* or *)
Quiet[Numerator @ CoefficientList[AsymptoticDSolveValue[{y[x]*y'[x]^2 == 2*(1y[x]), y[0] == 1}, y[x], {x, 0, 25}], x][[3;; 1;; 2]]] (* requires Mathematica 11.3+ *)


CROSSREFS

Cf. A070058, A093954, A202623, A335829 (denominators).
Sequence in context: A163775 A092244 A155634 * A003367 A121784 A213165
Adjacent sequences: A335825 A335826 A335827 * A335829 A335830 A335831


KEYWORD

nonn,frac


AUTHOR

Amiram Eldar, Jun 25 2020


STATUS

approved



