what is the name of this combinatorics formula? Jaynes gives the following formula:



Does this formula have a name? where can I find a justification for this formula? and what does the $\frac d {dx}$ mean on the right hand side? Does it mean the $n$'th derivative? because that would be astonishing to me.
 A: I don't know that this has a name.  However it follows from standard results in the theory of generating functions.  In particular the binomial series (https://en.wikipedia.org/wiki/Binomial_series) gives you
$$ (1-x)^{-(a+1)} = \sum_{k=0}^\infty (-x)^k {-(a+1) \choose k}$$
and you want to rewrite that binomial coefficient as one with positive arguments.  In this case
$$ {-(a+1) \choose k} = {(-(a+1))(-(a+2)) \cdots (-(a+k)) \over k!} = (-1)^k {(a+1)(a+2) \cdots (a+k)) \over k!} = (-1)^k {a+k \choose k}$$
so you get
$$ (1-x)^{-(a+1)} = \sum_{k=0}^\infty {a+k \choose k} x^k.$$
Now it is a standard result in the theory of generating functions that if 
$$ A(z) = \sum_{k \ge 0} a_k z^k $$
then
$$ z A^\prime(z) = \sum_{k \ge 0} k a_k z^k$$
which you can see by manipulating the power series.  Therefore, iterating this "take the derivative and multiple by $z$" operation $n$ times you get
$$ \left( z {d \over dz} \right)^n A(z) = \sum_{k \ge 0} k^n a_k z^k.$$  
Putting this identity together with the series for $(1+x)^{-(a+1)}$ I gave above gives the result.
For a friendly introduction to generating functions I like Wilf's generatingfunctionology.
