I have an integer sequence that I am trying to replicate. The formula in the OEIS page is as follows:
$$ \sum_{n,k} T(n,k) \frac{x^n}{n!} y^k = 1+\log\left(\sum_0^\infty\left((1+y)^\binom{n}{2}\frac{x^n}{n!}\right)\right) $$
I know what n and k in the formula are, however, I am not sure what $x$ and $y$ mean. What is $x$ and $y$?
This is a generating function. (I believe OEIS should have marked it as such; many OEIS entries have indications "g.f." for "generating function", or "e.g.f." for "exponential generating function.)
What that means is that if the function on the right side of the equals sign is expanded in powers of $x$, then the coefficient of $x^n$ will be a polynomial in $y$, and the coefficient of $y^k$ in that polynomial will be $\frac{T(n,k)}{n!}$. The sequence you want consists of the values of $T(n,k)$, so the coefficients in your expansion will have to be multiplied by $n!$. (The $n!$ in the denominator makes this an exponential generating function, rather than simply a generating function.)