If I have a list of numbers that isn't in OEIS, how would I go about determining if there's a relationship between these numbers in the form of a closed-form expression or nonlinear recurrence?
Some ideas that might help:
Find the first order differences, then the second order differences ... as many times as possible, depending on how many numbers you actually have, and look for a constant column. If you get a constant column of differences after some stage, then you have a polynomial.
If that fails, then try looking at the ratios of successive numbers. If they seem to be converging to a fixed real number, or appear to exhibit some pattern, such as oscillating, then you have something which might be something like a Fibonacci/Lucas sequence with a formula of the type $a^n+b^n$. If the ratios of consecutive terms are increasing at an approximately constant rate, then you might have something like a factorial - or even a gamma function.
Alternatively, post the numbers here, and see if anyone on MSE can suggest a relationship.