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Say that we have a sequence defined by the m

I'm curious as to why this formula involves binomial coefficients. Any ideas? Also. How could I prove this formula? With induction?

^{th}degree polynomial, ##a_n=\displaystyle \sum_{k=0}^{m}c_kn^k##. I found the following formula which is a recursive representation of the same sequence: ##\displaystyle a_n =\sum_{k=1}^{m+1}\binom{m+1}{k} (-1)^{k-1}a_{n-k}##.I'm curious as to why this formula involves binomial coefficients. Any ideas? Also. How could I prove this formula? With induction?

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