Consider the generating formula for the Bessel polynomials (increasing coefficients):
$$a_k=\frac{(N+k)!}{2^kk!(N-k)!}$$
I was trying to generate as high an order as possible with unsigned int ($2^{32}-1=4294967295$) in C++. The theoretical maximum should be for $N=10$ when $a_N=654729075$ (for $N=11$, $a_N=13749310575>4294967295$). I realize there are multiplications before divisions, so some higher value must be reached first before the final result, but I tried to expand the products and manually simplify the formula (a "spot the pattern" game), and I got to this point:
$$a_k=\frac{\prod_{i=max(N-k,k)+1}^{N+k}{i}}{2^k\prod_{i=1}^{min(k,N-k)}{i}}$$
Which works until $N=8$. This can be simplified even more considering that the denominator has powers of 2 that can be completely eliminated, and the rest of the terms which will end up perfectly dividing all the terms in the numerator. This leaves a simple product, but which doesn't have quite so recognizable patterns. Here is what wxMaxima outputs for $N=10$ (aligned to $11$, I don't know how to add spaces to have a match for every index):
$$\begin{align} & 1 \\\\ 5\cdot&11 \\\\ 3\cdot5\cdot9\cdot&11 \\\\ 2^2\cdot5\cdot9\cdot&11\cdot13 \\\\ 5\cdot7^2\cdot9\cdot&11\cdot13 \\\\ 3\cdot7^2\cdot9\cdot&11\cdot13\cdot15 \\\\ 2^2\cdot5\cdot7^2\cdot9\cdot&11\cdot13\cdot15 \\\\ 2^3\cdot5\cdot7\cdot9\cdot&11\cdot13\cdot15\cdot17 \\\\ 3\cdot5\cdot7\cdot9^2\cdot&11\cdot13\cdot15\cdot17 \\\\ 3\cdot5\cdot7\cdot9\cdot&11\cdot13\cdot15\cdot17\cdot19 \\\\ 3\cdot5\cdot7\cdot9\cdot&11\cdot13\cdot15\cdot17\cdot19 \end{align} $$
It's a crippled product of odd numbers, some repeating, and some even number squeezing in; the residuals of simplification.
Since the above seemed too much to implement in programming, I tried a recursive approach:
$$\frac{a_{k+1}}{a_k}=\frac{(N-i)(N+1+i)}{2i+2}$$
Which goes until $N=9$. I realize this is a bit of a whim, since there are builtin higher precision ints, and, if not, there's gmp, or similar, but my question is: can the formula be simplified even more? I am referring to the generating, not the recursive formula. Also, no tables.
In my humble opinion, computing first $$b_k=\log(\Gamma(N+k+1))-k\log(2)-\log(\Gamma(k+1))-\log(\Gamma(N-k+1))$$ using the LnGamma(x) function could be a way.
Then, $a_k=e^{b_k}$.