Given a polynomial sequence of some degree, how can find the polynomial?

Question

Let's say we are given a sufficient number of terms of a polynomial Sequence, does there exist a formula such that it can find the general term of the sequence(the polynomial equation)? I know method Polynomial interpolation, but other than that?

Answer 1

If a sequence is a polynomial sequence (which, I assume, means that $a_n = p(n)$ for some polynomial), then its first difference $b_n = a_{n+1}-a_n$ is a polynomial sequence for a $p$ of lower degree. (you can prove this, and also use it to construct an algorithm).

Answer 2

Use the equation $\,f(x) = \sum_{k=0}^\infty \Delta^k\!f(0) {x \choose k}\,$ which is a finite sum of binomial coefficients if $\,f(x)\,$ is given as a polynomial sequence. The use of this formula is explained in great detail in my answer to MSE question 4049277 "Avoiding brute force: determining when a specific polynomial in $\mathbb{Q}[x]$ is an integer for any integer $x$".