The number con_n counts matchings X on {1,2,... ,2n}, which are partitions into n two-element blocks, such that the crossing graph of X is connected.  Similarly, cro_n counts matchings whose crossing graph has no isolated vertex. (If it has no edge, Catalan numbers arise.) We prove, using a more generally aplicable criterion, that the sequences con_n and cro_n are not P-recursive. On the other hand, we show that the residues of con_n and cro_n modulo any fixed power of 2 can be determined P-recursively. We consider also numbers sco_n of symmetric connected matchings. Unfortunately, their OGF satisfies a more complicated differential equation which we cannot handle.