Summation proof

[Albert Chan]

(09-13-2023 01:44 PM)John Keith Wrote:  To clarify my (rather rusty) recollection, the LHS reads "the derivative of COMB(n, k) where x = 0"? Also, can you explain the connection to Bernoulli numbers?

It is d/dn of comb(n,k). No, not evaluate at n=0. We are proving an identity.

Connection to Bernoulli number is simply the method of using falling factorial form.
B(m) = d/dn Σ(x^m, x = 0 .. n-1), evaluated at n = 0

Proving this thread identity, we did the same thing (except n=0 part): Σ, then d/dn

Once we recognized d/dn comb(n,k) connection, the proof is easy.
We can actually look it up, from wiki for Binomial coefficient

Quote:The first three columns are very simple; subsequent columns do not seem obvious to me.

k = 1 .. n. First column (k=0) is not used. Diagonal are harmonic numbers.

I don't know if coefficients have combinatorial meaning.