I have a diamondback bmx and am sure it is authentic however it doesn't have a serial number anywhere on the frame and certainly not where you would expect it to be ( between where the arm and the crankshaft attach.
Almost all bikes have a serial number on them.
Exceptions might be
- A home made bike
- A small bespoke framebuilder (rare)
- A repaired bike where the part that had the number was replaced
- A stolen bike, where someone has ground-off the number to inhibit identification.
A serial number is not overly useful, since you know the brand, hopefully the manufacturer has a system, or has records still, and lets you look them up. More on serials: What is the purpose of a serial number?