Question

What is the generating function for the generating sequence A = 1, 9, 25, 49,…?

(a) 1+(A-x^2)

(b) (1-A)-1/x

(c) (1-A)+1/x^2

(d) (A-x)/x^3

The question was posed to me in examination.

The question is from Discrete Probability topic in chapter Discrete Probability of Discrete Mathematics

Answer 1

Right option is (b) (1-A)-1/x

The explanation: The generating function for the sequence A. Using differencing:

A = 1 + 9x + 25x^2 + 49x^3 + ⋯(1)

−xA = 0 + x + 9x^2 + 25x^3 + 49x^4 + ⋯(2)

(1−x)A = 1 + 8x + 16x^2 + 24x^3 +⋯. Since 8x + 16x^2 + 24x^3 + ⋯ = (1-x)A-1 ⇒  8 + 16x + 24x^2 +…= (1-A)-1/x.