3 of 3 deleted 2 characters in body

05AB1E, 10 9 bytes

-1 byte thanks to Erik the Outgolfer

sF³gݨm*O

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Takes x as first argument, n as second and p in ascending order as third.

Explanation

sF³gݨm*O
s         # Forces the top two input arguments to get pushed and swaped on the stack
 F        # Do n times...
  ³        # Push the third input (the coefficients)
   g       # Get the length of that array...
    ݨ     # and create the range [0 ... length]
      m    # Take the result of the last iteration to these powers (it's just x for the first iteration)
       *   # Multiply the resuling array with the corresponding coefficients
         O # Sum the contents of the array
          # Implicit print