[seqfan] Easy Proof Elaborating A006139

[Brad Klee]

In the latest draft, https://oeis.org/draft/A006139, we write out
an integral generating function with it's associated differential
equation. The comment proposes another geometric
generating function, with even more differential structure.

If this comment about a Hamiltonian Surface needs proof (?),
we can change coordinates by (q,p)=(sqrt(x)*Q,sqrt(x)*P), with
(Q,P) = (sin(u),cos(u)). again we have a quartic/quadratic form,

y = 2*H = (q^2+p^2)*(1-4*q*(q-p)),
y = 2*H = x - Z*x^2;  Z = 4*Q*(Q-P),

which implicitly defines the time integrand,

dt = dx/dy*du = 1/Sqrt(1-4*Z*y)*du ,

also the first energy derivative of the abbreviated action.
The integral of "dt" is almost the same as the one given
for A006139, but generates another integer sequence:

1, 4, 48, 640, 9520, 149184 . . .

Is this sequence a duplicate to A006139?

My answer is no, as the second integral seems more likely to
be measured in a high precision laboratory. For example
compare A000984 and A002894.

Have a Nice Day,

Brad