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 ?
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.
