I ran into an interesting variant of Pascal's triangle in generating an expression for an alternating sum of falling factorials
$$ \sum_{i=0}^{a} (-1)^i (i)_k = k! T(a, k) $$
where $T(a, k)$ satisfies the basic binomial recurrence but with different initial conditions
$$ \begin{align} T(a, k) &= T(a-1, k) + T(a-1, k-1) \\ T(a, a) &= 1 \\ T(a, 0) &= (1 + (-1)^a)/2 \\ \end{align} $$
which looks like
$$ \left( \begin{array}{ccccccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 2 & 2 & 1 & 0 & 0 & 0 \\ 1 & 2 & 4 & 3 & 1 & 0 & 0 \\ 0 & 3 & 6 & 7 & 4 & 1 & 0 \\ 1 & 3 & 9 & 13 & 11 & 5 & 1 \\ \end{array} \right) $$
Is there a generic name for the different sequences satisfying this recurrence or for these variants of Pascal's triangle in general?