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A number also called a Zag Number giving the number of Even Alternating Permutations. The first few are 1, 2, 16, 272, 7936, ... (Sloane's A000182).
See also Alternating Permutation, Euler Zigzag Number, Secant Number
References
Knuth, D. E. and Buckholtz, T. J. ``Computation of Tangent, Euler, and Bernoulli Numbers.'' Math. Comput. 21, 663-688, 1967.
Sloane, N. J. A. Sequence
A000182/M2096
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.