By Doron Zeilberger
(Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger)
Posted: Aug. 25, 2010.
Last update of this webpage (but not article): Jan. 3, 2016.
One of the main uses of computers is to do statistical analysis of data. But, so far, the theory of statistics, and its noble mother, Probability theory, were all discovered and developed by lowly humans. No more! Computers can also develop probability theory (and statistics), and discover (and prove!) general theorems of much larger depth than those discovered by human-kind. Of course, at this time of writing, they still need these inferior humans to give them a head-start by teaching (i.e. programming) them to develop probability theory ab initio (only better!), but even this will soon be superfluous.
[Some browsers need the .txt extension here is the same package with .txt HISTABRUT.txt]
(Note: see Shalosh B. Ekhad's note.)
f(z,x)=x+z*(f(z,x)-f(z,0))^2 + z*diff(f(z,x),x)
Here x is the catalytic variable, and the main interest is the sequence of numbers P_n(0), in other words, the coefficients of f(z,0).
But this also raises the question of whether the random variable accounted for by the exponent of x, is asympotically normal. Using the first 300 terms, it seems that it is.
In order to crank-out this sequence we the maple code
yields the following input and output files
input file yields the output file
[Note: To just see Noam0, see here]
input file yields the output file
[Note: it also needs the Maple package Findrec
input file yields the output file .