(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.
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) );
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);