A permutation π =(π1,...,πn) is consecutive 123-avoiding if there is no index i such that πi < πi+1 < πi+2. Similarly, a permutation π is cyclically consecutive 123-avoiding if the indices are viewed modulo n. We consider a weighted enumeration problem of consecutive 123-avoiding permutations where we able to to determine an asymptotic expansion of the weighted enumeration. For the cyclically weighted enumeration we are able to determine an exact expression in terms of the Hurwitz zeta function. This yields an explicit combinatorial expression for the higher derivatives of the cotangent function.