Generating functions homework question 3 This is a Homework question
Determine the Closed Form generating function for the sequence $a_0,a_1,a_2...,$ where $a_n$ is the number of partitions of the non negative integer n into
a) even summands
b)distinct odd summands
c) summands that do not exceed 10 and occurs at most $3$ times
d) summands where $1$ occurs at most once, summand $2$ occurs at most twice, and in general summand $k$ occurs at most $k$ times. 
I am not looking for complete answer but perhaps a hint or some help on how to set it atleast one of these up. 
 A: Hints: 


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*a) In general, the closed form generating function for the partitions of a non-negative integer $n$ is given by $F(x) = (1+x+x^{2}+\cdots)(1+x^{2}+x^{4}+x^{6}+\cdots)(1+x^{3}+x^{6}+\cdots)\cdots$. (Can you see why this is?) Now, if we want to restrict this to even summands, we note that if we were to restrict $F(x)$ to include only parenthetical polynomials which contained terms of even degree, then we would restrict the corresponding sequence in $a_{n}$ to those with non-zero entries only for even indices, hence limiting the partition to even (not necessarily distinct) integers. Thus, the generating function $K(x)$ corresponding to a partition of $n$ into even summands is given by $K(x) = (1+x^{2}+x^{4}+x^{6}+\cdots)(1+x^{4}+x^{8}+\cdots)\cdots$.

*b) By the same reasoning as above, the generating function for partitioning $n$ into non necessarily distinct odd summands is $Q(x) = (1+x+x^{2}+\cdots)(1+x^{3}+x^{6}+\cdots)(1+x^{5}+x^{10}+\cdots)\cdots$. To make this distinct, we note that if we were
to restrict each parenthetical polynomial in the product to just its first two terms, we would restrict the corresponding sequence in $a_{n}$ to entries of just $0$ or $1$, hence limiting each partition to distinct integers. Thus, the generating function $N(x)$ is: $N(x) = (1 + x)(1 + x^3)(1 + x^5)\cdots$.
Can you use these suggestions to solve the other two?
