Euler Zigzag Number

The number of Alternating Permutations for $n$ elements is sometimes called an Euler zigzag number. Denote the number of Alternating Permutations on $n$ elements for which the first element is $k$ by $E(n,k)$. Then $E(1,1)=1$ and

$$E(n,k)=\cases{ 0 & for $k\geq n$\ or $k<1$\cr E(n,k+1)+E(n-1,n-k) & otherwise.\cr}$$

See also Alternating Permutation, Entringer Number, Secant Number, Tangent Number

References

Ruskey, F. ``Information of Alternating Permutations.'' http://sue.csc.uvic.ca/~cos/inf/perm/Alternating.html.

Sloane, N. J. A. Sequence A000111/M1492 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.