Necessary and sufficient condition for sequence to be representable by a polynomial with integer coefficients

Question

I have derived a lemma for solving olympiad problems about whether some sequence S_n (n from a contiguous set of integers WLOG containing 0) can be represented by a polynomial P(n) with integer coefficients. I would like to know if it is already well known. It is as follows:

We define Dⁱ_n (where i is a nonnegative integer), the i^th level difference of the sequence S_n, as follows:

D⁰_n = S_n

Dⁱ_n = D^i-1_n+1-D^i-1_n

If n is from a finite set of size m, we can only define up to the m-1^th level difference.

Then, there exists a polynomial P(n) with integer coefficients such that P(n) = S_n if and only if all the elements of the sequence Dⁱ are divisible by i!. If n is from an infinite set, we also require that there exists some j such that for all k > j, D^k consists only of zeroes.

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To prove this, note that all polynomials with integer coefficients P(n) can be written in the form Σ C_i*_nP_i (where C_i are integer coefficients) using the division algorithm, with the converse being clearly true.

Then, as _nP_i=0 for n from 0 to i-1 and _nP_i=i! for n = i, for S_n=_nP_i, D^j₀=0 for j from 0 to i-1 and Dⁱ₀=i!. By induction on the degree of polynomial, D^j₀=0 for j > i.

Using the property that the differences of the sum of sequences are equal to the sum of the differences of sequences, we obtain that for S_n = P(n) = Σ C_i*_nP_i, Dⁱ₀=C_i*i!, which is divisible by i!. Thus, we have proven the "only if" part of the statement. To prove the "if" part of the statement, we construct Σ C_i*_nP_i for a sequence S_n by obtaining C_i from dividing Dⁱ₀ by i!. Since a sequence is completely defined by a column of differences, S_n = Σ C_i*_nP_i, and we obtain a polynomial with integer coefficients by expanding it.

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Is this a well known lemma, and where can I find similar useful lemmas?