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Two other references to similar sums are Bruce C. Berndt and Boon Pin Yeap, Explicit evaluations and reciprocity theorems for finite trigonometric sums, Advances in Applied Mathematics Volume 29, Is …

Another very short proof: the left side is a polynomial in $X$ of degree $a+b$ that takes integer values when $X$ is an integer, so it can be expressed in the form of the right side. The identity is …

The first formula is due to Herbert Ryser, who derived it from an inclusion-exclusion formula for the permanent of a matrix. It can be found in his book Combinatorial Mathematics, Mathematical Associa …

In John Riordan's book Combinatorial Identities, page 21, is the formula $$\sum_{k=0}^n \binom{n}{k}(x+k)^k (y+n-k)^{n-k}= \sum_{k=0}^n \binom{n}{k} k!\, (x+y+n)^{n-k}.$$ (There is a typo in the for …

Here's a proof of the identity. Unfortunately, it doesn't show how anyone would come up with this formula. Like the proofs referred to in the comments, this proof is based on the beta integral $$\int …

$\def\sone#1#2{\left[#1\atop #2\right]} \def\stwo#1#2{\left\{#1\atop #2\right\}} $ These formulas can be proved by Lagrange interpolation, using the fact that $\stwo{n}{n-m}$ and $\sone{n}{n-m}$ are …

Thanks to the OEIS, I found a reference to almost the same formula: Marko V. Jarić and Joseph L. Birman, New algorithms for the Molien function, Journal of Mathematical Physics 18, 1456 (1977); doi: …

The distribution of descents and leaves in forests of rooted trees is symmetric. See http://www.combinatorics.org/ojs/index.php/eljc/article/view/v3i2r8/pdf.

The fact that the binary operation defined by (2) is associative can be found in the Wikipedia article on formal groups (section 2) http://en.wikipedia.org/wiki/Formal_group. Presumably one might find …

Is Theorem 6.4 of http://people.brandeis.edu/~gessel/homepage/papers/enum.pdf what you want?

Let $c(n,k) = |s(n,k)| = (-1)^{n-k} s(n,k)$. It is well known that for fixed $k$, $S(n+k,n)$ and $s(n+k,n)$ are polynomials in $n$ of degree $2k$ with leading coefficient $(2k-1)!!/(2k)!$. If we exten …

The formula in 2 is a very special case of a result of Richard Stanley's, though it certainly could be older. (I wouldn't be surprised if it can be found in MacMahon's work.) See, e.g., my paper A his …

In terms of hypergeometric series, the sum is $_3F_2(-n, 1+n, 1/2;1,3/2;1)$ and the identity is a special case of Saalschütz's theorem (also called the Pfaff-Saalschütz theorem), one of the standard h …

The binomial identity is the well-known Vandermonde's theorem $\sum_{j=0}^n \binom{a}{j}\binom{b}{n-j} = \binom{a+b}{n}$ with $a=-i-1$, $b=i-m-1$.

You can use the approach of my paper A factorization for formal Laurent series, Journal of Combinatorial Theory, Series A 28 (1980) 321-337. Although this problem is not considered in that paper, The …