Can confirm a match with a HP laptop power plug (and Dell, I know that they're physically compatible, though machines refuse to charge when the wrong brand adapter is used). OD is 7.4, ID 5.0mm.
RE the centre pin, on the Dell adapters (and I suspect HP as well) it is connected to a maxim one-wire EEPROM type chip to supply an ID and charger capability info to the laptop (and at the same time making it harder to replace the adapter with a generic or DIY equivalent, making Dell/HP more money supplying replacements...). The one-wire chip has a habit of dying (who would have guessed having a ~5V max tolerant pin right next to 19.5V could occasionally end in tears!), going back to my previous comment about Dell/HP making money, not even counting the enormous waste of having to discard a perfectly functional PSU becasue an ID chip fails...
If your original PSU also has a similar ID system then you can probably find out (multimeter in ohms mode between outer shell and centre pin will probably show a moderate to high resistance (but not open circuit) if there is something similar present. If so you may need to see if you can copy the info onto a new chip for your DIY adapter - there will certainly be info on the web about doing this for the laptop chargers. If you're lucky it might turn out to be a simple resistor ID or simply un-used.