A short proof of the pentagonal number theorem

In 1960 Leonhard Euler gave rigorous proof of an efficient calculation using the recurrence of partition numbers. Since the power of the variables in the recurrence is the pentagonal numbers, this theorem is called the pentagonal theorem, whose contribution to the calculation of integer partitions is shown in the formula: p(n) = p(n-1) + p(n-2) - p(n-5) - p(n-7) + etc. Until now the pentagonal theorem has been widely known and used as an old classical math theorem, and it is not hard to prove it. However, I think it can still enlighten me on my way to studying mathematics even if it has been proved many years earlier. Just as the old saying says, “Timeless classics.” I think the true classics will never be out-fashioned such as the pentagonal theorem. And my study objectives are to deduct the pentagonal theorem again and give some subjective interpretation. In my deduction of the pentagonal theorem, I put myself in a situation where there hasn’t been an efficient way to calculate partition numbers. And my study objectives are to solve this problem. On my way to gain formula 1, I use generating functions of integer partitions and combinatorial transformation to figure out the coefficients of the recurrence of the partition numbers. And after formula 1 is approached, I gave a subjective interpretation of the relation between the pentagonal numbers and formula 1. And this bijection between them is proved not only through calculating pentagonal numbers but also through combinatorial explanations. And through my study on the pentagonal theorem, I come to find that it is important to combine the problems’ formula deduction with its combinatorial meanings concerning problems in permutation and combination.