Proofs involving Catalan Numbers

[saubbie]

Homework Statement

I need to prove two things about the Catalan numbers. The first is that Cn is odd iff n=(2^k)-1 for some positive integer k.

The second is that given the matrix A defined by the rule a(i,j)=C(i+j), prove that det A=1. I have not covered determinants in my linear class yet, so I do not have any idea what to do with this one.

Homework Equations

The Attempt at a Solution

I am assuming that both proceed by induction. The first one seems to be more direct, but I am really lost as to how to proceed with both. Any help or suggestions is greatly appreciated. Thanks a lot!

 

[HallsofIvy]

I would start by writing down the definition of Catalan numbers!

 

[saubbie]

Sorry. The Catalan numbers are 1,1,2,5,14,42,... given by the formula Cn=(1/(n+1))*2n choose n. With 2n choose n equal to (2n)!/(n!n!). Sorry for the bad form, I don't have any math writing program.

 

[HallsofIvy]

That's
$$\frac{1}{n+1}\frac{(2n)!}{n!n!}$$
Click on that to see the code. You don't need a "math writing program" on your own computer.