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We give a direct and simple proof of Touchard's continued fraction, providean extension of it, and transform it into similar expansions related to Motzkinand Schroeder numbers. Another proof is then given that uses only induction. Weuse this machinery on two examples that appear in recent papers ofJosuat-Verges; with an additional parameter, these two can be treatedsimultaneously.

On Touchard's continued fraction and extensions: combinatorics-free, self-contained proofs