write Recursion as a Binomial Coefficient

Question

supposedly I have this recursive function : $$ f(n,n)= f(n-1,n) + f(n,n-1) + f(n-1,n-1) $$ with $$ f(1,1) = 1 $$ $$ f(x,0) = 0 $$ $$ f(0,x) = 0 $$ The goal is to write this as a sum of binomial coefficients, something like this: $$ \sum_{i=0}^{n-1} \binom{-}{-}\binom{-}{-} $$ I sat the entire day on it and I still cannot find a way to do it..

Answer 1

This is sequence https://oeis.org/A001850, the formula being the head line . See the comment by B. Cloitre of 2002.