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.

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

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 three-body problem in 1882. In Meissel's expansion the coefficients are c(n)/2^n.

References

Sudhir Ranjan Jain, Mechanics, Waves and Thermodynamics: An Example-based 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. 1-11.

Links

Amiram Eldar, Table of n, a(n) for n = 1..243

S. K. Foong, From Moon-fall to motions under inverse square laws, European journal of physics, Vol. 29, No. 5 (2008), pp. 987-1003, alternative link.

Jaak Peetre, Ernst Meissel and the Pythagorean problem - the Drei-Körper-Problem in the Nachlass Meissel, draft, 1997.

Formula

a(n) = numerator(c(n)), c(1) = 1/2, c(n) = (2 * Sum_{k=1..n-1} (n-k)*(2*n-2*k-1)*c(n-k)*c(k) - Sum_{m=2..n-1} (n-m)*(2*n-2*m-1)*c(n-m) * Sum_{k=1..m-1} c(m-k)*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*(1-y[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: A092244 A342830 A155634 * A003367 A121784 A213165

Adjacent sequences: A335825 A335826 A335827 * A335829 A335830 A335831

Keyword

nonn,frac

Author

Amiram Eldar, Jun 25 2020

Status

approved