Discrete Convolution confusion

I am confused by the definition of convolution:

given 2 series {1,1} and {1,1},

what is their convolution?

Do I perform as below, taking the product of values only when Row1 and Row2 overlap:

Step 1:

......1 1....

...1 1.........

-------------

......1

Step 2:

... 1 1...

... 1 1...

-------------

... 1+1 = 2

Step 3:

...1 1...

..... 1 1

----------------

..... 1

Total = 1+2+1=4

Hence, the convolution of {1,1} and {1,1} is 4.

But I have a textbook which stated:

If for all ,

then

where { } is defined as the convolution of sequences { } and { }.

If I use the value of the first example, k=2, then

If you can clear the air, I would very much appreciate it.