><>, 30 bytes
Assumes input is on the stack. Outputs either a 1 or 0.
1v
<+1~v!?(}:{:,2*+1::
;n={<
This was fun. I'm new to ><>, and I'd welcome any suggestions for golfing.
This starts with n=1, then continually increments n while n(n-1)/2 is less than the input number. Once the loop terminates, it prints 1 if n(n-1)/2 is equal to the input number, 0 otherwise.