My favorite integer sequence

[sheldonison]

Then \( \rho \) has zeros where f(n) crisscrosses the A(n) partition function. Here is a possible estimate of the zero count of rho:
\( \text{zerocount}(\rho(\ln(x))) \approx 2 \frac{d}{dx}\ln(f(\exp(x))) \)

\( \text{zerocount}(f(-\ln(x))) \approx \frac{d}{dx}\ln(f(\exp(x))) \)

Uh ? can't we use d/dx ln(g(x)) = g ' (x) / g(x) ?

And further d/dx ln(g(exp(x)) = g ' (exp(x)) exp(x) / g(exp(x))

then since your LHS includes a ln I think we can substitute ln(x) = y or y = exp(x) and get

zerocount(rho(x)) ~~ 2 f ' (x) / f(x)

and zerocount(rho(-x)) ~~ f ' (x) / f(x)

and similar for the half-iterate of exp.

Im a bit tired so forgive if my calculus is bad today.

regards

tommy1729

Hey Tommy, I think your calculus is correct! Very nice. That simplifies the heck out of the equations .... by the way, that should be h(n) from the other post... kinda busy today and tomorrow morning though.