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$A(x)$ is generating function of $\{a_n\}_{n=0}^{\infty}$ and $B(x)$ is generating function of $\{b_n\}_{n=0}^{\infty}$, what is $$A(x^2)+xB(x^2)$$?

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Hint: we have $A(x^2) = \sum_{n=0}^\infty a_nx^{2n} = a_0 + a_1x^2 + a_2x^4+ \cdots$. What can we say about the coefficients of odd powers ($x, x^3, \dots$)?

Now do the same for $xB(x^2)$, what can we say about the its even power coefficients?

What does this mean for their sum?

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  • $\begingroup$ $A(x^2)=\frac{1}{1-x^2}$? $\endgroup$ Mar 25 at 13:07
  • $\begingroup$ @john $A(x^2)$ is the generating function of the sequence $a_0,0,a_1,0,a_2,0,...$, the sequence that alternates values of the sequence $a_n$ and zeros. $\endgroup$
    – plop
    Mar 25 at 13:30
  • $\begingroup$ Thanks it's amazing @plop. $\endgroup$ Mar 25 at 14:41

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