The On-Line Encylopedia of Integer Sequences

So I am finally getting into my summer routine, which means that I will be able to blog at least every 2-3 days, but hopefully more often...

Regardless, while doing some combinatorics on trees I was reminded of the most excellent website brought to you by N. J. A. Sloane at AT&T Research. (Who knew phone companies still had labs?) Anyway, pretty much any sequence you can think of is already up there and it tells you many of the known applications of the sequence. And if you know of a sequence that is not up there or of an application of a sequence that is already in the database, then you can e-mail the webmaster and get it on there.

For example, I have been dealing with a sequence that I call the "phylogenetic numbers" since they show up in the study of phylogeny, i.e. evolutionary trees. So I put the first part of the sequence into the database (1, 3, 15, 105, 945) and it returned this, which are (as it turns out) the double factorial numbers of 2n-1, i.e. (2n-1)!!. And then they give about a dozen applications of this sequence.

I have to warn you, that this site can consume hours and hours of your time. Just for fun, you should look at the Catalan numbers. Speaking of number sequences and combinatorics, I suggest that you get familiar with the two-volume opus by Prof Stanley on Enumerative Combinatorics.

Addendum
I realized that I should state briefly what the double-factorial numbers are. Two references are here and here. The gist of the double factorial numbers is the following.

n!!=(n-2)(n-4)(n-6)...kn; where kn is 1 if n is odd and 2 if n is even. E.g., 8!!= 8*6*4*2 and 9!!=9*7*5*3*1