The following is an addition to 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 ?