> One of the (sets of) functions haven't yet found a suitable open source 
alternative
> for is related to guessing formulas and generating functions for an input 
sequence
> (as in Mathematica's FindSequenceFunction and FindGeneratingFunction).

I am amazed that no one has yet mentioned Gfun, the classic Maple package 
for 
the manipulation of generating functions by Bruno Salvy and others which is 
released under the GNU Lesser General Public License (LGPL v2.1). 

http://perso.ens-lyon.fr/bruno.salvy/software/the-gfun-package/
http://www.swmath.org/software/344
https://scm.gforge.inria.fr/anonscm/gitweb/?p=gfun/gfun.git;a=tree

Peter
 

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