I need some help figuring out the closed form of the following expansion.
T[n]=T[n-1]+T[1]*T[n-2]+T[2]*T[n-3]+T[3]*T[n-4]+...+T[n-1]
I haven't done this type of math in a while. I am solving a programming problem and this turns out to be the answer to the program. I am just curious to find out if it has a closed form in mathematics. It the tags I specify are inaccurate, please feel free to add others.
This equation of reccurence is well known. It defines the Catalan numbers. There is a closed form for the $n$-th term of the sequence in term of a binomial coefficient: $$ T[n] = \frac{1}{n+1}\binom{2n}{n} $$