The partition numbers

are 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, …. They count the number of ways of splitting an integer

into a sum of positive integers, without regard to order. For instance,

, the number of ways to partition 4, is 5 because there are 5 possible sums that add to 4: 4, 1+3, 2+2, 1+1+2, 1+1+1+1. By convention,

.

The left-hand side of the identity shows the product of geometric series

with common ratios

,

,

,…,

, each truncated to

terms. The right-hand side multiplies them out and collects terms.

As an example, let

; the term

on the right is the sum of the five products

,

,

,

,

, where the factors in each product are in the order of the series they come from. These products correspond to the partitions 4, 1+3, 2+2, 1+1+2, 1+1+1+1.

In general, an uncollected term in the expansion of the left side is of the form

. The factor

comes from the

term of the

truncated series

and contributes

(

times) in the corresponding partition. Collecting terms amounts to counting the number of ways

can be partitioned.