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 $\displaystyle a_k = b_k = 1$ for all $\displaystyle k\geq0$,

then $\displaystyle c_k=a_k*b_k = k+1$

where {$\displaystyle c_k$} is defined as the convolution of sequences {$\displaystyle a_k$} and {$\displaystyle {b_k}$}.

If I use the value of the first example, k=2, then $\displaystyle c_k=3$

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