3 a tiny misprint corrected

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 A000111 gives the formula $$A_m=2^n\biggl|E_m\biggl(\frac12\biggr)+E_m(1)\biggr| A_m=2^m\biggl|E_m\biggl(\frac12\biggr)+E_m(1)\biggr|$$ 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.

2 Toss to Ross, correct unless known nickname

Dear TossRoss,

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.

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.