Which one of the following is the closed form for the generating function of…

GATE · 2022 · CS · Computer Science & IT

Which one of the following is the closed form for the generating function of the sequence {an}n≥0\{a_n\}_{n\ge 0} defined below?

an={n+1,n is odd1,otherwisea_n=\begin{cases}n+1,&n\text{ is odd}\\1,&\text{otherwise}\end{cases}

  1. A.

    x(1+x2)(1−x2)2+11−x\frac {x(1 + x^2)} {(1 - x^2)^2} + \frac {1} {1 - x}

  2. B.

    x(3−x2)(1−x2)2+11−x\frac {x(3 - x^2)} {(1 - x^2)^2} + \frac {1} {1 - x}

  3. C.

    2x(1−x2)2+11−x\frac {2x} {(1 - x^2)^2} + \frac {1} {1 - x}

  4. D.

    x(1−x2)2+11−x\frac {x} {(1 - x^2)^2} + \frac {1} {1 - x}

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