Explicit formula for Euler zigzag numbers(Up/down numbers)

ross_tang

I have derived an explicit formula for the Euler zigzag numbers, the number of alternating permutations for n elements:
[tex] A_j=i^{j+1}\sum _{n=1}^{j+1} \sum _{k=0}^n \frac{C_k^n(n-2k)^{j+1}(-1)^k}{2^ni^nn} [/tex]

For details, please refer to my article in Voofie:

An Explicit Formula for the Euler zigzag numbers (Up/down numbers) from power series

I would like to ask, if my formula is new, or is it a well known result? Since I can't find it in Wikipedia or MathWorld. If it is an old formula, can anyone give me some reference to it?

arildno

Well, mathworld mentions something called the "Entringer numbers", in its article:
http://mathworld.wolfram.com/EulerZigzagNumber.html
It doesn't seem to be quite the same thing??

CRGreathouse

There are many formulas at Sloane's A000111 though I don't see yours, at least directly. But it may be there in disguised form.

Computationally, your formula is not competitive with some of the others listed there. For example, given

your code A(500) takes four times longer than Somos' code a(500). I don't know if yet more efficient code exists.

ross_tang

@arildno
Thank you for the link. I haven't read the "Entringer numbers" before. But I don't found explicit formula, and seems the Euler zigzag number is it's special case.

@CRGreathouse
I have read the link you send me before. Thank you. First, I found out the formula, not because it is computationally efficient. I am interested in the explicit form of it. Since the zigzag number sequence is solution to a few number of recurrence relation. And the Somo's code you mentioned used recurrence relation too. But I found none explicit form in the link.