Why is $\binom{a}{n}=(-1)^{a}\frac{\sin(n\pi)}{(a+1)\binom{n}{a+1}\pi}$? Why is $\displaystyle\binom{a}{n}=(-1)^{a}\frac{\sin(n\pi)}{(a+1)\binom{n}{a+1}\pi}$?
(A particular case popped up as an alternative formulation in WolframAlpha while operating with binomials. Any explanation for their relationship with sine function over pi times a polynomial?)
EDIT: This function only gets integer values different from zero for $\{0,1,\ldots,a-1,a\}$. Certainly, for the mentioned set  $\binom{n}{a+1}=0$, so the equality will only be defined for $n\geqslant a+1$, where it will be 0 for all integers due to $sin(n\pi)$.
Ex: For $\binom{3}{n}$

 A: It's not "really" the sine function, just a convenient way to keep track of signs, sort of like writing $ \cos (n \pi) $ when $ n $ is an integer, instead of $(-1)^n $.
Now, $\sin (n \pi) = 0$ for $ n $ an integer. Here it looks like $n $ is a half integer and $\sin (n pi)$ is plus or minus one according to whether the numerator is 1 or 3 mod 4.
Sorry for terse explanation, on a tablet.
A: This is a perfect example of the kind of nonsense that results from expanding binomial coefficients into factorials and then factorials into applications of the Gamma function, without regard to the cases where one gets undefined expressions. In most cases when one writes a binomial coefficient$~\binom nk$, one has in mind integers $n,k$ with $k\geq 0$ and most often $k\leq n$. Now if you assume $0\leq n\leq a$ integers, then you perfectly defined and integer left hand side is equated to $(-1)^a\frac0{0\pi}$ (the $0$ in the denominator is due to the binomial coefficient $\binom n{a+1}$ being zero when $a+1>n\geq0$ are integers). This is undefined in any case, and has the smell of being non-integer due to the factor$~\pi$ in the denominator, but of course it cannot really be said to be so. In case $n>a$ instead, the left hand side is zero and $\binom n{a+1}>0$ so the right had side becomes zero as well; at least this one entirely uninteresting case makes sense.
There is probably some delicate sense in which by taking limits the right hand side can be massaged into something meaningful. However this requires a lot of care and the wrong way of taking limits may result in the wrong (not just underfined) values. If you really want to make sense of this I suggest you read section 5.5 (Hypergeometric functions) of Concrete Mathematics to the end; page 216 has a discussion of taking limits, which although I did not succeed in 
understanding it, looks as a warning against a naive approach to such expressions.
Personally I find that the otherwise nice subject of binomial coefficient identities looses its appeal when attacked with hypergeometric techniques. Unfortunately CA systems seem to adore hypergeometrics, and more then once I have found that they will throw hypergeometric functions at you on occasions where there is no need for them at all. Try entering the Vandermonde convolution into WA, and appreciate the result (which aught to be a single binmial coefficient).
