Test your intuition: consecutive tails

Something very unfortunate has happened: several things I have recently written that could have been blog entries are instead answers on math.SE! In the interest of exposition beyond the Q&A format I am going to “rescue” one of these answers. It is an answer to the following question, which I would like you to test [...]

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Extracting the diagonal

Needless to say, I have been very, very busy. But enough about me. Suppose you are given a bivariate generating function in “closed form,” where I’ll be vague about what that means. Such a generating function may arise, for example, from counting lattice paths in ; then might count the number of paths from to [...]

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The Catalan numbers, regular languages, and orthogonal polynomials

I’ve been inspired by The Unapologetic Mathematician (and his pages and pages of archives!) to post more often, at least for the remainder of the summer. So here is a circle of ideas I’ve been playing with for some time. Let be the ordinary generating function for the ordered rooted trees on vertices (essentially we [...]

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