Some recurrence relations in finite topologies

In a number of papers (see, e.g., RZhMat, 1977, 11B586) there is given for the number $T_0(n)$ of labeled topologies on $n$ points satisfying the $T_0$ separation axiom the formula
$$ T_0(n)=\sum\dfrac{n!}{p_1!\dots p_k!}V(p_1,\dots,p_k), $$
where the summation extends over all ordered sets $(p_1,\dots,p_k)$ of natural numbers such that $p_1+\dots+p_k=n$. In the present paper there is found a relation for calculating, when $n\geqslant2$, the sum of all terms in this formula for which $p_2=1$ in terms of the values $V(q_1,\dots,q_t)$ with $q_1+\dots+q_t\leqslant n-2$. This permits the determination (with the aid of a computer) of the new value
$$ T_0(12)=414\,864\,951\,055\,853\,499. $$