Consider the sequence of integers defined recursively by $c_0 = 1$ and $$ c_p = \sum_{l = 0}^{p-1} (-1)^{p+l+1} \binom{p}{l}^2 c_l $$ for $p \geq 1$. This is sequence A000275 in the online encyclopedia of integer sequences and its first few members are 1, 1, 3, 19, 211, 3651.
I believe this sequence comes up in a problem I'm dealing with (calculating the norm of a $(p,p)$-form on an $n$-dimensional complex vector space) but I can't quite prove that the numbers I have are the $c_p$. I've got it down to showing that if a number $A_{n,p}$ (for $n$ and $p$ such that $2p \leq n$) satisfies $$ \binom{n}{p} \Biggl( A_{n,p} + \sum_{l=0}^{p-1} (-1)^{p+l} c_l \binom{p}{l}^2 \cdot \frac{\binom{n-p+l}{p-l}}{\binom{n}{p-l}} \Biggr) = 1 $$ then $A_{n,p} = c_p$ (unless I've made a mistake somewhere). But here I'm stuck. I'm awful with all things involving finite sums and binomial coefficients, can someone help here?