Dear Toss,
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 A000111 gives the formula $$ A_m=2^n\biggl|E_m\biggl(\frac12\biggr)+E_m(1)\biggr| $$ where $E_m(x)$ are the Euler polynomials 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