Series Solution of Non-Linear Equation

This sequence (formed by the convolution of preceeding terms) reminds 
me of the sequence of coefficients for the power series solution of 
the equation 
                  x x''  +  a (x')^2  =  b                     (1)

Among the solutions of this equation (with appropriate choices of a,b)
are exp(t), sin(t), cos(t), (A+Bt)^n, A+Bt+Ct^2, and sqrt(A+Bt+Ct^2).
This last function represents the separation between any two objects 
in unaccelerated motion.  Other solutions include the cycloid relation
for (non-rotating) gravitational free-fall, and the radial distance
of a mass from a central point about which it revolves with constant 
angular velocity and radial freedom.

The power series solution of equation (1) can be written

     x(t)  =  c[0]  +  c[1] t  +  c[2] t^2  +  c[3] t^3 + ...

where the coefficients c[i] satisfy the convolutions

                 n                               /  b  if n = 2
               SUM  A(k,n-k) c[k] c[n-k]    =   (
                k=0                              \  0  if n > 2

with
            A(k,j) = (a-1) j (k-j) + k(k-1)/2

Any choice of c[0], c[1], c[2], and c[3], with c[1]c[2] not zero, 
determines the values of a,b and therefore all the remaining 
coefficients.  There are many interesting things about these sequences
of c[k] values.  Focusing on just the sequences with |c[k]| = 1,
k=0,1,2,3, there are obviously 16 possible choices, but only 8 up to
a simple sign change.  These 8 can be arranged as four groups of 2:

  k          I                II             III           IV
 ---      --------        ----------      ---------     ---------
  0       1      1        -1      1        1     1       1     1
  1       1     -1         1      1        1    -1       1    -1
  2       1      1         1     -1       -1    -1       1     1
  3      -1      1         1      1       -1     1       1    -1
  4      1/2    1/2       3/2   -3/2       0     0       1     1
  5      1/2   -1/2       5/2    5/2      4/5  -4/5      1    -1
  6     -3/2   -3/2       9/2   -9/2      2/5   2/5      1     1
  7      3/2   -3/2      19/2   19/2     -2/5   2/5      1    -1
  8      3/8    3/8     133/8 -133/8     -1/2  -1/2      1     1
  9    -29/8   29/8     267/8  267/8     1/30 -1/30      1    -1
                                     etc

Clearly the coefficients in each group differ only in sign.  The
coefficients in groups I and II diverge, and those in group IV are
all units.  Only the group III sequences converge.  Interestingly,
these coefficients are given very closely by

                 c[k-1]  =  2 exp(uk) sin(wk)

for k>2, where
                     u = -0.145370157...

                       /  1.877672951...  for III(a)
                  w = (       
                       \  1.263919649...  for III(b)

Notice that the two possible values of w sum to 3.1415926...

The integer numerators and denominators of these c[k] sequences also
have many interesting properties.  For example, primes p congruent to 
+1 (mod 4) first appear in the denominator at c[p], whereas primes 
congruent to -1 (mod 4) first appear at c[p^2].  The sequence of
numerators is much less regular

 1  -1  -1  1  0  -4  2  2  -1  -1  59  -9  -1  233  8  -934  49 .. etc

Incidentally, the value of b in the ubiquitous equation (1) is 
essentially just a constant of integration, and the underlying
relation is the derivitive

                     x x'' + q x' x''  =  0

where q=3 for unaccelerated separations and q=2 for (non-rotating)
gravitational separations.  Isolating q and differentiating again
leads to the basic relation, free of arbitrary constants,

 x x' x'' x'''' - x x' (x'')^2 - x (x'')^2 x''' + (x')^2 x'' x''' = 0

Dividing by x x' x'' x''' gives the nice form

           x''''      x'''      x''       x'
          ------  -  -----  -  ----  +  -----  =   0
           x'''       x''       x'        x