Final answer:
The generating functions of the number kn of partitions of n which contain only perfect squares can be computed using series expansion and the binomial theorem. By applying a specific process, the generating function can be simplified to 2n2.
Step-by-step explanation:
The generating functions of the number kn of partitions of n which contain only perfect squares can be computed using a method called series expansion. One approach is to use the binomial theorem, which states that (a + b)n is equal to an + n*an-1*b + n(n-1)*an-2*b2/2! + ...
In the case of partitions of n that contain only perfect squares, imagine taking (n - 1) from the last term and adding it to the first term. This process can be repeated until you reach the second term, where you take (n - 3) from the penultimate term and add it to the second term. Ultimately, this leads to a generating function of 2n2 for the number of partitions.
For example, if we want to compute the generating function for n = 25, we start with 12 + 32 + ... + (2*25 - 3)2 + 252. Following the process described above, we get 2*252 = 1250 as the generating function.