I have an infinite set of polynomials, each of which is comes from higher order term in the series expansion of some function about zero. I seek a two-variable generating function that will spit out the coefficients of these polynomials (one variable for the order of the expansion we are looking in, and another for the term in the polynomial we want the coefficient of).
I have a Mathematica program that can spit out arbitrarily many terms in the expansion from which I can read off the coefficients, but I am wondering if there is some pattern to determine them generally outside of the program. The first few coefficients follow for example, with $\alpha$ being the order in the series expansion.
$$ \alpha=0: \{1\}\\ \alpha=1: \{2,1\}\\ \alpha=2: \{6,4,\frac{1}{2}\}\\ \alpha=3: \{20,15,3,\frac{1}{6}\}\\ \alpha=4: \{70,56,14,\frac{4}{3},\frac{1}{24}\}\\ \alpha=5: \{252,210,60,\frac{15}{2},\frac{5}{12},\frac{1}{120}\}\\ etc... $$
There is a pattern in the first AND last number of each series. If we call $\delta$ the index of the elements of each set, with $\delta\in[\alpha,2\alpha]$, then this pattern in the first/last coefficients has been identified as
$$ C_{\alpha,\delta}=\frac{(2-\delta_{\alpha,0})}{\delta!} \prod_{\chi=2}^{\alpha}(4\chi-2)\quad,for~\delta=\alpha,2\alpha $$
where above, $\delta_{\alpha,0}$ is the Kronecker delta. Clearly, each series of coefficients is decreasing monotonically as $\delta$ increments, but not by any global pattern that I can identify.
The original function is
$$ \frac{1}{\sqrt{1-s^2\Omega}}\exp{\left(\frac{\sqrt{1-s^2\Omega}-1}{2s^2Q}+\frac{Q}{4\Omega}\right)} $$ which gets expanded about $s=0$, then brought into my form with the replacement $\Omega\rightarrow4\Lambda$. The coefficients then are of increasing powers of $\Lambda$, with the values of $\delta$ having a 1-1 correspondence to the powers of $\Lambda$ in a given term.
Could someone help me determine a generating function for these coefficients?
I was close with the first and final coefficients. The middle sets are filled in with application of the Binomial coefficients:
$$ C_{\alpha,\delta}=\frac{(2-\delta_{\alpha,0})}{\delta!} \prod_{\chi=2}^{\alpha}(4\chi-2){\alpha\choose{\delta-\alpha}} $$
This generates all the coefficients correctly.