Summary: Recently, \textit{T. Kløve} [ibid. 41, 298-300 (1995; Zbl 0828.94019)] analyzed the average worst case probability of undetected error for linear \([n, k; q]\) codes of length \(n\) and dimension \(k\) over an alphabet of size \(q\). The following sum: \[ S_n= \sum^n_{i=1} {n \choose i} \Biggl( {i\over n} \Biggr)^i \Biggl( {{1-i} \over n} \Biggr)^{n-i} \] arose, which has also some other applications in coding theory, average case analysis of algorithms, and combinatorics. Kløve conjectured an asymptotic expansion of this sum, and we prove its enhanced version in this correspondence. Furthermore, we consider a more challenging sum arising in the upper bound of the average worst case probability of undetected error over systematic codes derived by \textit{J. Massey} [Proc. Int. Conf. Inf. Theory Syst. (Berlin, 1978), 307-315]. Namely, \[ S_{n,k}= \sum^n_{i=1} {{n-k} \choose i} \Biggl( {i\over n} \Biggr)^i \Biggl( {{1-i} \over n} \Biggr)^{n-i} \] for \(k\geq 0\). We obtain an asymptotic expansion of \(S_{n,k}\), and this leads to the conclusion that Massey's bound on the average worst case probability over all systematic codes is better for every \(k\) than the corresponding Kløve bound over all codes \([n, k; q]\). The technique used in this correspondence belongs to the analytical analysis of algorithms and is based on some enumeration of trees, singularity analysis, Lagrange's inversion formula, and Ramanujan's identities. In fact, \(S_n\) turns out to be related to the so-called Ramanujan's \(Q\)-function which finds many applications (e.g., hashing with linear probing, the birthday paradox problem, random mappings, caching, memory conflicts, etc.).