The following is an addition to http://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) =$q^3+2 q^2+6 q+6$ ; $q^{t_{1}}$ = $q^1$  ; coeff. =6;  
f(n=3)=14 and $c_{4}$(q) =$q^6+2 q^5+6 q^4+14 q^3+22 q^2+36 q+24$ ;  $q^{t_{2}}$ = $q^3$ ; 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 ?