0, 0, 0, 1, 2, 3, 4, 5, 6, 7,……………..

Assume the generating function

$f(x)=a_0+a_1x+a_2 x^2+a_3 x^3+……$

But, the given sequence is {0, 0, 0, 1, 2, 3, 4……}. Using this sequence, the expression above becomes

f(x)$=0 +0x+0x^2+1x^3+2x^4+3x^5+4x^6+…… \\ =1x^3+2x^4+3x^5+4x^6+…… \\ =x^3 (1+2x+3x^2+4x^3+…….) \\ =x^3 (1-x)^{-2}$

Accordingly, $f(x) = x^3 (1-x)^{-2}$ is the generating function for the given sequence {0, 0, 0, 1, 2, 3, 4, 5, 6, 7,……………..}

6, -6, 6, -6, 6, -6, 6, ……………………..

Assume the generating function

$f(x)=a_0+a_1x+a_2 x^2+a_3 x^3+……$

But, the given sequence is {6, -6, 6, -6, 6, -6, 6, …………………}. Using this sequence, the expression above becomes

f(x)$=6 -6x+6x^2-6x^3+6x^4…… \\ =6(1 -x+x^2-x^3+x^4……) \\ =6(1+x)^{-1}$

Accordingly, $f(x) = 6(1+x)^{-1}$ is the generating function for the given sequence {6, -6, 6, -6, 6, -6, 6, ……………………..}