Ring generated by binomial polynomials

Definition
Let $$R$$ be a commutative unital ring of characteristic zero. Let $$K$$ be the ring obtained by localizing $$R$$ at the multiplicative subset of nonzero integers. Then, the ring generated by binomial polynomials over $$R$$ is the subring of $$K[x]$$ comprising all $$R$$-linear combinations of the polynomials:

$$\binom{x}{r} = \frac{x(x-1)(x-2) \dots (x - r + 1)}{r!}$$.

where $$r \ge 0$$ (for $$r = 0$$, this is the constant polynomial $$1$$).

Equivalently, it is the tensor product with $$R$$ of the ring generated by binomial polynomials over the rational integers, i.e., the ring generated by binomial polynomials over $$\mathbb{Z}$$.

Equivalently, it is the ring $$\operatorname{Int}(\mathbb{Z},R)$$: the ring of all polynomials $$f \in K[x]$$ such that $$f(\mathbb{Z}) \subseteq R$$.

Facts

 * Ring of integer-valued polynomials is contained in ring generated by binomial polynomials
 * Ring of integer-valued polynomials over rational integers equals ring generated by binomial polynomials