  The discrete-time quantum walk is a quantum counterpart of the random walk.
It is expected that the model plays important roles in the quantum field. In
the quantum information theory, entanglement is a key resource. We use the von
Neumann entropy to measure the entanglement between the coin and the particle's
position of the quantum walks. Also we deal with the Shannon entropy which is
an important quantity in the information theory. In this paper, we show limits
of the von Neumann entropy and the Shannon entropy of the quantum walks on the
one dimensional lattice starting from the origin defined by arbitrary coin and
initial state. In order to derive these limits, we use the path counting method
which is a combinatorial method for computing probability amplitude.
