Generalised Pascal Triangle with two-periodic formula

Question

Consider a generalisation of pascal's triangle of the following form $$1 \\ 1 \ 1 \\ 1\ 2\\ 1\ 3\ 2\\ 1\ 4\ 5\\ 1\ 5\ 9\ 5$$ That is to say, for even-numbered rows the $n$-th term is found by adding the $(n-1)$-th and $n$-th terms of the previous row, and for odd-numbered rows, the rule is similar, except that no new term is added on the end at all. (For example, there is no $1$ on the end of the third row above.)

Is it possible to find a explicit expression for the entries of this triangle?

Motivation: This problem is related to the following problem of enumerative geometry: 'Let $n\geq 2$, given $(n-2)$-dimensional subspaces $H_1,\ldots, H_{2(n-2)}$ of $\mathbb{P}^n$ how many lines are there, in general, intersecting all spaces?’

Answer 1

If you start with Pascal's triangle

1
1  1
1  2  1
1  3  3  1
1  4  6  4  1
1  5 10 10  5  1
1  6 15 20 15  6  1    

precede and follow with $0$s

0  1  0
0  1  1  0
0  1  2  1  0
0  1  3  3  1  0
0  1  4  6  4  1  0
0  1  5 10 10  5  1  0
0  1  6 15 20 15  6  1  0    

take differences between consecutive terms

1 -1
1  0 -1
1  1 -1 -1
1  2  0 -2 -1
1  3  2 -2 -3 -1 
1  4  5  0 -5 -4 -1
1  5  9  5 -5 -9 -5 -1

and finally drop the zero and negative values in the right-hand half

1 
1 
1  1 
1  2 
1  3  2  
1  4  5 
1  5  9  5 

you get your triangle.

So, one possibility for a formula is $${n \choose m}-{n \choose m-1}$$ assuming you start counting rows and columns that way starting from $0$. For example the $9$ here is ${6 \choose 2}-{6 \choose 1} = 15-6$

Answer 2

As the comment by Semiclassical indicates, this is highly relevant to Catalan's triangle, defined by the recurrences $C(n+1,k)=C(n+1,k-1)+C(n,k)$ for $1<k<n+1$, $C(n+1,n+1)=C(n+1,n)$ for $n\geq1$, and the boundary conditions $C(n,0)=1$ for $n\geq0$ and $C(n,1)=n$ for $n\geq1$. There is an explicit formula for $C(n,k)$ given by $$C(n,k)=\frac{(n+k)!(n-k+1)}{k!(n+1)!}.$$ Now, if the $k$th entry in the $n$th row of your triangle is called $D(n,k)$, then it is easy to notice that $D(n,k)=C(n-k+1,k-1).$ (You can convince yourself by looking at how $D(n,k)$ are entries along the rising diagonal on the table of values provided in the Wikipedia page, or by verifying the defining recurrence is satisfied like so.) Consequently we have $$\boxed{D(n,k)=C(n-k+1,k-1)=\frac{n!(n-2k+3)}{(k-1)!(n-k+2)!}}.$$