Find the coefficient of `alpha^(6)` in the product `(1+alpha+alpha^(2))(1+alpha+alpha^(2))(1+alpha+alpha^(2)+alpha^(3))` `(1+alpha)(1+alpha)(1+alpha)`

The given product can be written as
`(1+alpha+alpha^(2))(1+alpha+alpha^(2))(1+alpha+alpha^(2)+alpha^(3))(1+alpha)^(3)`
or `(1+alpha+alpha^(2))(1+alpha+alpha^(2))(1+alpha+alpha^(2)+alpha^(3))(1+3alpha+3alpha^(2)+alpha^(3))`
Multiplying Synthetically: `{:(1,alpha,alpha^(2),alpha^(3),alpha^(4),alpha^(5),alpha^(6), . . .),(1,3,3,1,0,0,0,):}`
. .. on multiplying by `1+alpha+alpha^(2)+alpha^(3)to`To each coefficient add 3 preceding coefficient
`(:(1,4,7,8,7,4,1, . .):}`
. . . on multiplying by `1+alpha+alpha^(2)to`To each coefficient add 2 preceding coefficient.
`{:(1,5,12,19,22,19,12, . . .):}`
. . . on multiplying by `1+alpha+alpha^(2)to` toeach coefficient add 2 preceding coefficient.
`{:( . . . ., . . ., . . , . . ., . .. ,53, . .):}`
Hence, required coefficient is 53.