I'm reading the definition of the Witt ring in N. Jacobson's book "algebra II", section 8.10. What I will write is in page 502. In it he defines the maps $\phi$ and $\psi$ which are inverse of one to the other in the ring $X^m$, $X$ a ring. So $\phi$ takes "regular" coordinates and gives "ghost" coordinates and $\psi$ goes the other way.
For $\psi$ he writes that:
Let $A=(a^{(0)},...,a^{(m-1)})$ and define $\psi A$ as $(a_0,...,a_{m-1})$ where: $$a_0=a^{(0)},$$ $$a_\nu=\frac{1}{p^\nu}(a^{(\nu)}-a_0^{p^\nu}-pa_1^{p^{\nu-1}}-...-p^{\nu-1}a_{\nu-1}).$$ This is, what is visible in my copy. I tried to write $a_1$ and when it looked a bit strange, I tried to write the recuring formula which seems to be the following: $$a_\nu=\frac{1}{p^\nu}(a^{(\nu)}-\sum_{i=0}^{\nu-1}p^{i}a_i^{p^{\nu-i}}).$$ Then I noticed that the last term of this sum is $p^{\nu-1}a_i^p$ which is not the same with Jacobson's last term, i.e. $p^{\nu-1}a_{\nu-1}$.
So I'm asking if someone can write the closed formula for that term and apologies if the whole question is wrong, due to something invisible in my copy of the book.
Since I found the answer and there was no reply, I'm posting it to close the thread.
Jacobson has made a typo. The formula (48) in page 502 needs to be as in the following screenshot.
It is easy to see why it is correct and also Hasse in the number theory book pg. 157 of the english version writes the formula for the first term, i.e. $a_1.$