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Thomas Baruchel's mathematical notebook

Thomas Baruchel's mathematical notebook

(https://gist.github.com/2b870728323456f71402)

This is my mathematical notebook; I put here the identities I find from time to time; this notebook is written in the markdown format with the vim editor. I use my own vim-notebook plugin for embedding Maxima code in it; thus I can evaluate each cell of code.

I am a contributor to the OEIS and some of the identities below may be found there also.

A formula discovered in august 2014:

I found this formula after having written an article with Jean-Paul Allouche: (http://arxiv.org/abs/1408.2206)

/*                            oo                 n      */
/*                    5      ====   (1 - n)  (-1)       */
/*                           \                          */
/*      erf(1)  =  -------    >     --------------      */
/*                     ___   /                          */
/*                 2 \/ pi   ====    (2n - 5) n!        */
/*                           n=0                        */

/* (Maxima code) */
float(erf(1));
float(5/(2*sqrt(%pi))*sum( (1-n)*(-1)^n / ((2*n-5)*n!),n,0,256) );

An identity concerning sequence A108626

While working with Prof. Dr. Carsten Elsner on some properties of constant ζ(2) to be published, I spent much time studying the sequence A108626 (itself related to sequence A108625) from the OEIS.

See my contributions in A108626.

See my contributions in A171155 that I discovered to be related to A108626.

Here is a recurrence relation:

/*    A108626(n+2) - 2*A108626(n+1) - A108626(n) =         */
/*                                                         */
/*          n      k                                       */
/*         ====   ====                                     */
/*         \      \     ( n+1-k ) ( n+1-k ) ( n+1-i )      */
/*      2   >      >    (       ) (       ) (       )      */
/*         /      /     (  i-1  ) (   i   ) (  k-i  )      */
/*         ====   ====                                     */
/*         k=0    i=0                                      */

/* (Maxima code) */
a(n) := sum(sum(
  binomial(n-k, i)^2 * binomial(n-i, k-i),
  i, 0, k), k, 0, n);
makelist(a(n+2)-2*a(n+1)-a(n), n, 0, 12);
makelist(2*sum(sum(
   binomial(n+1-k, i-1) 
 * binomial(n+1-k, i)
 * binomial(n+1-i, k-i),
 i, 0, k), k, 0, n), n, 0, 12);
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