As of August 27, 2009, Stan Wagon informed the author, quoting Herbert Wilf, that the following proof is well known and published in the book by S. K. Lando,
Lectures on Generating Functions, Providence, RI: AMS, 2003.
The formal generating function of Euler numbers
has the simple continued fraction
;
likewise, the formal generating function for the absolute value of the Euler numbers
is
.
Hence, the formal generating functions of the sequences
and
have a terminating continued fraction expansion, and thus are rational functions in
. It is well known that a rational generating function gives rise to a periodic sequence in modular arithmetic.
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