gaussian kernel and bionomial coefficients
The well-known twentieth-century American columnist Dorothy Parker once reviewed a book 
saying 


  "This is not a book to be put down lightly; it should be thrown with great force."


Be that as it may, Professor Collins is suggesting that a bar plot of the binomial
coefficients on the $n$-th row of Pacal's triangle, after suitable normalization,
will look from far away like a Gaussian distribution with mean $n/2$ and standard
deviation $\sqrt{n}/2$.  He then further obfuscates this by using the fact that for
even values of $n$, we can write the standard deviation as $\sqrt{k/2}$
where $k = n/2$
while for odd $n$, it is simply $\sqrt{n}/2$, inviting the reader to arrive at
the mistaken conclusion that there is a fundamental difference between the odd
and even rows.

To get a better understanding of where the $n/2$ and $\sqrt{n}/2$
comes from, look for reading material on the DeMoivre-LaPlace approximation
which is a special case of the Central Limit Theorem.