Some recurrence relations in finite topologies

In a number of papers (see, e.g., RZhMat, 1977, 11B586) there is given for the number T_o(n) of labeled topologies on n points satisfying the T_o separation axiom the formula $$T_0 (n) = sum {frac{{n!}}{{p_1 ! ldots p_k !}}V(p_1 , ldots p_k ),} $$ where the summation extends over all ordered sets (p₁,...,P_k) of natural numbers such that p₁+...+P_k=n. In the present paper there is found a relation for calculating, whenn?2, the sum of all terms in this formula for which p₂=1 in terms of the values V(q₁,...,q_t) withq₁+...+q_t?n-2. This permits the determination (with the aid of a computer) of the new value T_o(12)=414 864 951 055 853 499