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  
d/dx g(x, q)  == g(x, q)^2 * g(q* x, q),  
with series g(x,q)==sum( k=0..$\infty$; $c_{k}$(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( { | $(\lambda)$ | = n };  $f(\lambda)$  ) ;  

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