Since Jess Tauber brought up recently the subject on the seqfan mailing list recently here are two variations on +Harlan Brothers
' idea for finding a relation between the binomial triangle and the constant e
A) Let's consider the limit of the n*(n-1)/2 -th root of the product of all binomial coefficients of order n.
(see first formula image)
It seems to converge slowly to a value close (?) to e.
B) Let's consider the limit of the n*(n+1)*(n-1)/6-th root of the product of all binomial coefficients (n choose i) raised at the i-th power.
(see second formula image)
It seems to converge slowly to another value greater than 4.
In each case the idea is to count the number of factors incorporating the variable n in the product and then applying the corresponding inverse power.
The correct offset for the polynomials involved in the exponentials might be different from the one I suggest for best results.