Recurrence relation for partition function with sum of divisors

Theorem

The following formula holds: $$p(n)=\dfrac{1}{n}\displaystyle\sum_{k=1}^n \sigma_1(k) p(n-k),$$ where $p$ denotes the partition function and $\sigma_1$ denotes the sum of divisors function.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $24.2.1 \mathrm{II}.A.$