Finding a polynomial of order h from the first h+1 terms

Question

In class we were finding a quadratic series from the first couple terms, I found that the equation of a quadratic sequence p₁, p₂, p₃ is (p₁-2p₂+p₃)x^2/2 +(-5p₁+8p₂-3p₃)x/2 +3p₁-3p₂+1p₃ but I want to make an equation that works for any order polynomial. The method to solve them regularly for quadratics is to find the difference between each term and continue down until a row is constant (Picture), I think I found an equation for the c^th set of terms t_n in row k

$$\sum_{n=1}^{h+1}\left(\sum_{t=0}^{k}\frac{\left(-1\right)^{\left(t+k\right)}k!}{\left(k-t\right)!t!}\left(c+t\right)^{n}\right)t_{n}$$

I have no clue if it is correct honestly as I'm not really used to using sum notation and such questions. I would rather confirm I at least have this part correct before ! continue to solve for t_n in terms of p_n. Also the letters I chose for certain variables might be confusing and I know that. I hope someone can enlighten me with their wisdom

edit: I'm aware of method of differences as that is the original method I used but I'm trying to find it without finding all the differences manually but find a formula using only the first few terms.