Depressed Equations and Generalized Catalan Numbers

Discriminating Deltas, Depressed Equations, and Generalized Catalan Numbers is a set of notes on the the relation of generating functions of the generalized Catalan numbers, e.g., OEIS-A001764, to the compositional inverse of G(x,t)= x + t \: x^n and the tangent envelope of associated discriminant curves.

Added 2/2014: See also “Polygonal Dissections and Reversions of Series” by Alison Schuetz and Gwyn Whieldon for relations between the generalized Catalan (Fuss-Catalan) numbers and dissections of polygons.

The original title was “Discriminating Deltas, Depressed Equations, and Fussy Catalan Numbers,” but some Russian (Vlad the Impaler) with an inferiority-superiority complex took umbrage at this play of words at the expense of a fellow comrade and made a fuss about it.

Related Stuff:

MSE-Q: Taylor series of the inverse of x^4 + x. Note how  that the limit of convergence of the series is determined by the discriminant curve, as noted in my comments there and apparent from the  graphical presentation in my paper.

MO-Q: Compute inverse series for implicit equation. Note that many of the fomulas in the pdf are not restricted to integer n and apply to n>0.

 

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One Response to Depressed Equations and Generalized Catalan Numbers

  1. Tom Copeland says:

    See also pages 59-62 of Huygens and Barrow, Newton and Hooke by Arnold.

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