elliptic curves - Alternate expresion of L-series coefficients

There are two different recursions involved here, one for the points of $E$ over ${mathbb F}_{p^n}$, and the other for the coefficients of the $L$-function.

If we write $a_p = alpha + beta,$ where $alphabeta = p$ (so $alpha$ and $beta$ are
the two roots of the char. poly. of Frobenius), then

$$1 + p^n - E({mathbb F}_{p^n}) = alpha^n + beta^n.$$

On the other hand, the Euler factor at $p$ for the $L$-function of $E$ is
$$(1 - alpha p^{-s})^{-1}(1-beta p^{-s})^{-1}$$
$$= (1 + alpha p^{-s} + alpha^2 p^{-2s} + cdots )(1 + beta p^{-s} + beta^2 p^{-2s} +
cdots )$$
$$= 1 + (alpha + beta) p^{-s} + (alpha^2 + alphabeta + beta^2) p^{-2s} +
cdots ,$$
and so we conclude that $a_{p^n}$ (the coefficient of $p^{-ns}$ in the $L$-function)
equals
$$alpha^n + alpha^{n-1} beta + cdots + alphabeta^{n-1} + beta^n.$$

These formulas are simply different, as soon as $n > 1.$ The recursion given in
the question describes the second, and not the first.