I wonder if the function $(1+y)(1+y^2)(1+y^3)(1+y^4)\cdots, 0< y<1$, converges to some well-known function.
If we let $ (1+y)(1+y^2)(1+y^3)(1+y^4)\cdots = \prod_{i=1}^\infty (1+y^i) = \sum_{i=0}^\infty a_i y^i$ then $a_i$ satisfies the following relation: $$a_0=1, i > 0$$ $a_i$ is the the number of partitions $(b_1, \cdots , b_s) $ of natural number $i$ such that $\sum_{t=1}^s b_t =i$ and $b_t < b_{t+1}$. For instance $a_{10} = 9$
$$(1,2,3,4) \; (1,2,7) \; (1,3,6) \; (1,4,5) \; (1,9) \; (2,8) \; (3,7) \; (4,6) \; (10)$$
$a_{10}$ is like Ramanujan's function $p(n)$. Is there anything in number theory related with $a_i$? At any rate the infinite product converges the function we know well? That is to say, it is a
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I found the material :
Let $p(n) $ is a Ramanujan function. For instance p(4) =4 :
1+1+1+1 = 1+ 3 =2+2 = 2+ 1+1
$\sum^\infty_{n=0} p(n) x^n = \Pi^\infty_{n=1} \frac{1}{1-x^n}$
Here the convenience is $p(0)=0$