#Mathematica, 49 bytes
f@2=4;f@1=0;f@n_:=Tr@Range@f[n-1]-Tr@Array[f,n-1]
Defines function f. 1-indexed.
#Explanation
f@2=4;f@1=0;f@n_:=Tr@Range@f[n-1]-Tr@Array[f,n-1]
f@2=4;f@1=0; (* Define f[1] and f[2] *)
f@n_:= (* f[n] equals ... *)
Tr@Range@f[n-1] (* Sum of (1, 2, ..., f[n-1]) *)
-Tr@Array[f,n-1] (* Minus the sum of previous terms *)