Recurrences for the partition function and its relatives (Q1880824)

Two new recurrences are given for the unrestricted partition function \(p(n)\), one of which is more efficient than Euler's pentagonal number recurrence. Recurrence formulas are also given in terms of \(p(n)\) for the function \(q(n)\), representing the number of partitions of \(n\) into distinct parts, and for the function \(q_0(n)\), representing the number of partitions of \(n\) into distinct odd parts.