Editing Binomial coefficient (section)

=== Integer-valued polynomials ===
{{main|Integer-valued polynomial}}
Each polynomial <math>\tbinom{t}{k}</math> is [[integer-valued polynomial|integer-valued]]: it has an integer value at all integer inputs <math>t</math>. (One way to prove this is by induction on ''k'' using [[Pascal's identity]].) Therefore, any integer linear combination of binomial coefficient polynomials is integer-valued too. Conversely, ({{EquationNote|4}}) shows that any integer-valued polynomial is an integer linear combination of these binomial coefficient polynomials. More generally, for any subring ''R'' of a characteristic 0 field ''K'', a polynomial in ''K''[''t''] takes values in ''R'' at all integers if and only if it is an ''R''-linear combination of binomial coefficient polynomials.