Aspire 5735z power cycling

I have an Aspire 5735z that has gradually developed an odd habit: it power cycles if I plug it in when it's off. As long as it's only connected to power when it's on, it does fine. If I forget and it does start looping, the power button light and comes on, the fan starts whirring, but I don't believe the hard drive spins up, and then after a couple of seconds it shuts off and tries again. It never posts. I can't find anything on twenty minutes' Googling to quite match, but some information does indicate a possible mobo problem?

The computer is about two years old; it was a floor model at Best Buy, and for most of its life it stayed connected to power and rarely ran off battery. It still has all stock hardware, but I've upgraded to Win7 and it has two flavors of Linux and Haiku installed as well. Never updated the BIOS. It has had one significant overheating incident that I'm aware of, although it seemed to do well enough, and I don't know how hot it actually got.

Does anyone know why connecting the power would cause cycling like that? It doesn't matter if I unplug it again before I turn it on, as best I've been able to tell. Just having it plugged in at any point is enough to cause the issue.

TIA for any help!