[math-fun] one-line/proof without words/elegant proof that coeffs of (1+x)^n approx a Gaussian?

Is there a simple/elegant/one-line proof that the coefficients of (1+x)^n approximate a Gaussian curve? I know that it is the convolution of n boxes, and that a large convolution of non-impulses approaches a Gaussian, but is there a quick/obvious proof of this that also gives you the appropriate mean & std deviation parameters?