What will be the sequence generated by the generating function 4x/(1-x)2

To find the sequence generated by the generating function 4x/(1-x)^2, we can use the concept of generating functions.

Step 1: Rewrite the generating function in a simplified form. In this case, we have 4x/(1-x)^2.

Step 2: Expand the denominator using the binomial theorem. The binomial theorem states that (1+x)^n = 1 + nx + (n(n-1)/2!)x^2 + (n(n-1)(n-2)/3!)x^3 + ...

In our case, we have (1-x)^2 = 1 - 2x + x^2.

Step 3: Substitute the expanded denominator back into the generating function. We have 4x/(1-x)^2 = 4x/(1 - 2x + x^2).

Step 4: Multiply both the numerator and denominator by (1-x)^2 to simplify the expression. We get 4x(1-x)^2/(1 - 2x + x^2)(1-x)^2.

Step 5: Simplify the expression further. We have 4x(1-2x+x^2)/(1 - 2x + x^2 - 2x + 4x^2 - 2x^3 + x^2 - 2x^3 + x^4).

Step 6: Combine like terms in the numerator and denominator. We have 4x - 8x^2 + 4x^3 / 1 - 4x + 6x^2 - 4x^3 + x^4.

Step 7: The coefficients of the terms in the generating function represent the terms of the sequence. So, the sequence generated by the generating function 4x/(1-x)^2 is 4, -8, 4, 0, 0, ...