Recursion series with two variable indices $u_{n,k} = u_{n-1,k} + u_{n-2,k-1}$

Question

I found a recursive formula that looks very similar to Pascal's triangle. Where in Pascal's triangle you have the recurring formula

$$u_{n,k} = u_{n-1,k} + u_{n-1,k-1},$$

which gives the Binomial function $\binom{n}{k}$ with the initial condition $u_{1,0}=u_{1,1} = 1$. In my case the second term is one value lower in the index

$$u_{n,k} = u_{n-1,k} + u_{n-2,k-1} , $$

with the initial condition $u_{1,0} = u_{2,0} = 1 $ and $ u_{2,1} = 2$.

It looks like the image above (where we miss the top 1). Where n is the height and k is the width. How can one calculate a closed form for this? I've done this bedore with single variable recursion formulas by using the characteristic polynomial, but I'm not sure how to start now.

Answer 1

Hint The entries in the $k$th antidiagonal are binomial coefficients $n \choose k$.