The following is an addition to https://mathoverflow.net/questions/132338/a-function-from-partitions-to-natural-numbers-is-it-familiar; the function $f(\lambda)$ as defined there, when summed over all partitions of n, gives an unexpected hit in  http://oeis.org/A232434. Checked up to n=36. 

In short:  
Define  $g(x, q)$ by  $$\frac{\partial g(x,q)}{\partial x}=g(x,q)^2 * g(q x,q),$$ 
with series $$g(x,q)=\sum _{k=0}^{\infty } \frac{ c_k(q) x^k}{k!},$$ 
define  $t_{n}=n(n+1)/2$; then the coefficient of $q^{t_{n-1}}$ in $c_{n+1}(q)$ is 
 $\sum{_{\lambda \in n}}{f(\lambda)}.$ 

examples:  
f(n=2)=6 and $c_{3}(q) =q^3+2 q^2+6 q+6$ ; $q^{t_{1}} = q^1$ with coeff. =6.  
Likewise f(n=3)=14 and $c_{4}(q) =q^6+2 q^5+6 q^4+14 q^3+22 q^2+36 q+24$ ;  
$q^{t_{2}} = q^3$ with coeff. =14;  

Remark that the differential equation has no closed form solution, but its series can be generated upto $x^n$ for any given n.  

Question: are there any arguments to trust\distrust this conjecture? Can it be proved ?

correction 28/05/2014 22:02 CET:  the differntial equation has a product in the RHS, not a sum. Thanks to Pietro for spotting it.