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Well...

sage: (LHS(n,k)-RHS(n,k)).simplify_factorial().canonicalize_radical()
0

is a big fat red hint...

Well...

sage: (LHS(n,k)-RHS(n,k)).simplify_factorial().canonicalize_radical()
0

is a big fat red hint...

EDIT : The big fat red hint is that canonicalize_radical transforms an "horrible`expression to someting more palatable...