Which of the following is the closed form of the ordinary generating function…

Which of the following is the closed form of the ordinary generating function
G(x)=∑n=0∞anxnG(x)=\sum_{n=0}^{\infty} a_nx^n for the sequence {an}n≥0\{a_n\}_{n\geq 0}
defined below?
an=n+1a_n=n+1, if nn is odd; otherwise, an=1a_n=1.

  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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