You are very generous. I have you to thank for resolving this. Thank you.

I think I may know what caused the problem. Usually, I set the notebook's default context to "unique to this notebook". I left one notebook in global context. That notebook had a different formula for p(x) than the one in the new notebook.

Try a fresh restart of Mathematica to clear the hidden cache of previous work and then

In[1]:= Clear["`*"];
p0[i_]:=Sum[(-1)^j 2^(2j-2)(Binomial[i+j-2,2j-2](2i-1)/(2j-1)),{j,1,i}]
p[i_,x_]:=Sum[(-1)^j 2^(2j-2)(Binomial[i+j-2,2j-2](2i-1)/(2j-1))x^(2j-1),{j,1,i}]
p0[3]
p[3,1]

Out[4]= -1

Out[5]= -1

Check this very carefully to make certain I haven't made any mistake.

And then the hard part, see if you can remember and reproduce what you might have done to get the different result and figure out why that gave you the different result. IF you can successfully do that then that might help protect you in the future when it isn't as obvious that a result isn't what it should be. That is what the outcome of this really should be. I hope it works.