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By analogy with Stirling numbers, tri-restricted numbers of the second kind count the number of partitions of a given set into a given number of parts, each part being restricted to at most three elements. Tri-restricted numbers of the first kind are then defined as elements of the matrix inverse to the matrix of tri-restricted numbers of the second kind. A new recurrence relation for the tri-restricted numbers of the second kind is presented, with a combinatorial proof. In answer to a problem that has remained open for several years, it is then shown by determinantal techniques that the tri-restricted numbers of the first kind also satisfy a recurrence relation. This relation is used to obtain a reciprocity theorem connecting the two kinds of tri-restricted number.