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We study a recurrence equation of type \[ l_n \left(2^{n + s} - 2\right) = 2^n a_n + \sum_{k = 1}^{n - 1} \begin{matrix} n \\ k \end{matrix} l_k ,\quad n\geqq N \] where $a_n $ is any sequence and $s,N$ are integers. This type of recurrence arises in many applications in computer sciences and telecommunications, e.g., in the analysis of unsuccessful search in a family of Patricia trees, in the average complexity of an algorithm generating exponentially distributed variates, in trie statistics, in the performance evaluation of conflict resolution algorithms in a broadcast communication environment, etc. We present a closed-form solution of the recurrence and then we establish an asymptotic approximation for it. In addition, we offer an approximation of a generating function, $l(z)$, of $l_n $ for small values of z.

Solution of a Linear Recurrence Equation Arising in the Analysis of Some Algorithms