8 Aug
2013
8 Aug
'13
5:52 p.m.
On 08/08/2013 06:42, James Propp wrote:
Here's something I posted to the domino forum back in 1996:
The recurrence relation C_n = ((4n-2)/(n+1)) C_{n-1} for the Catalan numbers can be turned backwards into the "precurrence" C_{n-1} = ((n+1)/(4n-2)) C_{n}. This formula eventually gives us zero (when n+1 vanishes) and gives us zero forever afterwards, but interestingly, with its "dying breath" the formula confides to us that the negative first Catalan number is -1/2.
Is this just deathbed raving, or does this actually mean something?
Well, it simplifies the generating function from (1 - sqrt(1-4x)) / 2x to -sqrt(1-4x) / 2x. I'm not sure that's much more than raving, though. -- g