There's a key difference between a null array and an empty array. This is a test for null.

int arr[] = null;
if (arr == null) {
  System.out.println("array is null");
}

"Empty" here has no official meaning. I'm choosing to define empty as having 0 elements:

arr = new int[0];
if (arr.length == 0) {
  System.out.println("array is empty");
}

An alternative definition of "empty" is if all the elements are null:

Object arr[] = new Object[10];
boolean empty = true;
for (int i=0; i<arr.length; i++) {
  if (arr[i] != null) {
    empty = false;
    break;
  }
}

or

Object arr[] = new Object[10];
boolean empty = true;
for (Object ob : arr) {
  if (ob != null) {
    empty = false;
    break;
  }
}

They're two completely different things. == compares the object reference, if any, contained by a variable. .equals() checks to see if two objects are equal according to their contract for what equality means. It's entirely possible for two distinct object instances to be "equal" according to their contract. And then there's the minor detail that since equals is a method, if you try to invoke it on a null reference, you'll get a NullPointerException.

For instance:

class Foo {
    private int data;

    Foo(int d) {
        this.data = d;
    }

    @Override
    public boolean equals(Object other) {
        if (other == null || other.getClass() != this.getClass()) {
           return false;
        }
        return ((Foo)other).data == this.data;
    }

    /* In a real class, you'd override `hashCode` here as well */
}

Foo f1 = new Foo(5);
Foo f2 = new Foo(5);
System.out.println(f1 == f2);
// outputs false, they're distinct object instances

System.out.println(f1.equals(f2));
// outputs true, they're "equal" according to their definition

Foo f3 = null;
System.out.println(f3 == null);
// outputs true, `f3` doesn't have any object reference assigned to it

System.out.println(f3.equals(null));
// Throws a NullPointerException, you can't dereference `f3`, it doesn't refer to anything

System.out.println(f1.equals(f3));
// Outputs false, since `f1` is a valid instance but `f3` is null,
// so one of the first checks inside the `Foo#equals` method will
// disallow the equality because it sees that `other` == null

array1.equals(array2) is the same as array1 == array2, i.e. is it the same array. As @alf points out it's not what most people expect.

Arrays.equals(array1, array2) compares the contents of the arrays.

Similarly array.toString() may not be very useful and you need to use Arrays.toString(array).

First up, yes, you're right that the second null check is redundant. If obj is null then the method will return false on the first check:

    if (!(obj instanceof ArrayOfColumn)) return false;

Of course, that would better be written:

    if (!(obj instanceof ArrayOfColumn)) {
        return false;
    }

But, that's the least of my concerns. The synchronized equals is probably there because other methods are synchronized, but synchronization comes at a cost. Unless you are sure you need it, remove it.

Additionally, synchronized methods are seldom the best solution. It is normally better to have tighter control of your locks so that nobody but you can hold them. That normally means using a private, internal "monitor" for synchronization:

private final Objct lock = new Object();

private boolean equals(Object obj) {
    synchronized(lock) {
        .....
    }
}

Using the private lock prevents other people from hanging your application by using your class as their own monitor. Imagine if your class is called MyClass, and someone does:

 private MyClass instance = new MyClass();

 synchronized(instance) {
     Thread.sleep(10000);
 }

If they did the above, your methods in other threads would never run, and your application would "hang".

Now, abut the __equalsCalc variable. It does nothing useful. It is set, and then cleared, inside the synchronized method, and, as a result, it will never, ever have any value in it that is useful. It is completely redundant and a waste of time.

Oh, and what's with the fully qualified java.lang.Object? Just use Object.

I would rewrite the method as (using the same synchronized method to be compatible with other methods in your class):

public synchronized boolean equals(Object obj) {
    if (!(obj instanceof ArrayOfColumn)) {
        return false;
    }
    if (obj == this) {
        return true;
    }
    ArrayOfColumn other = (ArrayOfColumn) obj;
    if (getColumn() == null) {
        return other.getColumn() == null;
    }
    return other.getColumn() != null && Arrays.equals(column, other.getColumn());
}

Note that you still have a synchronization vulnerability - the other column's column could be changed between checking for whether it's column is null, and using the column in the equals. You could still get null pointer exceptions if someone changes the other column in the middle of your equals. You should consider synchronizing on that other column too:

public synchronized boolean equals(Object obj) {
    if (!(obj instanceof ArrayOfColumn)) {
        return false;
    }
    if (obj == this) {
        return true;
    }
    ArrayOfColumn other = (ArrayOfColumn) obj;
    synchronized(other) {
        if (getColumn() == null) {
            return other.getColumn() == null;
        }
        return other.getColumn() != null && Arrays.equals(column, other.getColumn());
    }
}

Unfortunately, cross-synchronizing like that can lead to deadlocks if you are not careful.

No you can't remove an element from an array, as in making it shorter. Java arrays are fixed-size. You need to use an ArrayList for that.

If you set an element to null, the array will still have the same size, but with a null reference at that point.

 // Let's say a = [0,1,2,3,4] (Integer[])
 a[2] = null;
 // Now a = [0,1,null,3,4]

No.

An ArrayList can be empty (or with nulls as items) an not be null. It would be considered empty. You can check for am empty ArrayList with:

ArrayList arrList = new ArrayList();
if(arrList.isEmpty())
{
    // Do something with the empty list here.
}

Or if you want to create a method that checks for an ArrayList with only nulls:

public static Boolean ContainsAllNulls(ArrayList arrList)
{
    if(arrList != null)
    {
        for(object a : arrList)
            if(a != null) return false;
    }

    return true;
}

I generally use a static utility function that I wrote called equalsWithNulls to solve this issue:

Copyclass MyUtils {
  public static final boolean equalsWithNulls(Object a, Object b) {
    if (a==b) return true;
    if ((a==null)||(b==null)) return false;
    return a.equals(b);
  }
}

Usage:

Copyif (MyUtils.equalsWithNulls(s1,s2)) {
  // do stuff
}

Advantages of this approach:

• Wraps up the complexity of the full equality test in a single function call. I think this is much better than embedding a bunch of complex boolean tests in your code each time you do this. It's much less likely to lead to errors as a result.
• Makes your code more descriptive and hence easier to read.
• By explicitly mentioning the nulls in the method name, you convey to the reader that they should remember that one or both of the arguments might be null.
• Does the (a==b) test first (an optimisation which avoids the need to call a.equals(b) in the fairly common case that a and b are non-null but refer to exactly the same object)