Build a generating function for ar the number of integer sol

Build a generating function for ar, the number of integer solutions to the following:

e1+e2+e3+e4=r, 2<=ei<=7 e1 even, e2 odd

Solution

Build a generating function for ar, the number of integer solutions to the following:

e1+e2+e3+e4=r

and 2<=ei<=7 i=2,3,4,5,6,7 and e1 even; e2 odd

the genarating function:g(x):(x2+x4+x6)²(x3+x5+x7)²