 Single-block DNA zipper partition function

A homogeneous DNA zipper permitting exactly one contiguous paired block of length $j\ge 1$ has $N-j+1$ block positions and weight $\kappa s^{j-1}$. Consequently its partition function is $q=\kappa {\sum }_{j=1}^N(N-j+1)s^{j-1}=\kappa [s^{N+1}-(N+1)s+N]/(s-1{)}^2$. The removable value at $s=1$ is $\kappa N(N+1)/2$. At large s, $q\sim \kappa s^{N-1}$. Including an unpaired state gives the full partition function $1+q$; the large-s limit is not uniform in arbitrarily small initiation weight.