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I have recently needed to compute the number of permutations in $S_n$ with all cycles odd, and while this is easy using Flajolet-Sedgewick theory (the exponential generating function seems to be $\sqrt{\frac{1+z}{1-z}},$) I wonder who and when did it first (so I can reference this properly).

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See the OEIS,, in particular the reference to Riordan's book. – Ira Gessel Jul 5 '13 at 17:19
@IraGessel, shouldn't that be an answer? – François G. Dorais Jul 6 '13 at 1:03
It's not really an answer, just a suggestion. – Ira Gessel Jul 6 '13 at 15:48
In fact, I followed your suggestion and read Riordan. It is clear that the statement is easy once one reads his discussion (starting on p. 71), it is not actually there. It is also not clear who the results go back to (conjecturally, they go back to Cauchy). – Igor Rivin Jul 6 '13 at 17:06
It seems likely that the formula might appear in Pólya's 1937 paper (on "Pólya's theorem") but I haven't checked this. It's also always a good idea to check Richard Stanley's Enumerative Combinatorics for anything on enumerative combinatorics. – Ira Gessel Jul 7 '13 at 17:16

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