My favorite integer sequence

[tommy1729]

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