Dear Ross,

It looks that you don't really wish to see known formulae
for your zigzag numbers. Otherwise I don't understand why
you found my search insufficient.

The OEIS <a href="http://www.research.att.com/~njas/sequences/A000111">A000111</a> gives the formula
$$
A_m=2^n\biggl|E_m\biggl(\frac12\biggr)+E_m(1)\biggr|
$$
where $E_m(x)$ are the <a href="http://en.wikipedia.org/wiki/Bernoulli_polynomials">Euler polynomials</a>
which can be generated by the following explicit expansion
$$
E_m(x)=\sum_{n=0}^m\frac1{2^n}\sum_{k=0}^n(-1)^k\binom nk(x+k)^m,
$$
a double sum as in your case. Even if this formula is not exactly the same as yours
(although it looks pretty similar), this is a *known* double sum expression for $A_m$.
There is a lot of room for playing with this double sum and producing many other
(useful and useless) formulae for the zigzag numbers.

And don't forget: I've never seen this specific sequence before.

Best wishes,
Wadim