While semi-randomly browsing, I came across this conjecture which Philippe Flajolet sent to Doron Zeilberger as a "gift" (the "gift" is here, so you can check to see if I have typeset it correctly): http://www.math.rutgers.edu/~zeilberg/fp09.pdf ):
GIFT. Define the “equiharmonic numbers” by $$K_ν := \frac{(6ν)!}{Ω^{6ν}} \sum\limits_ {(n_1,n_2)∈(Z×Z)-{(0,0)}} \frac1{(n_1e^{−2iπ/3} + n_2e^{2iπ/3})^{6ν}} , Ω := \frac1{2π} Γ(1/3)^3 . $$
Show that the generating function of $(K_ν)$ admits the continued fraction representation $$\frac{7}{36} \sum\limits_{ν≥1} K_νz^{ν−1} = \frac{1}{1 − \dfrac{d_1 · z}{1 − \dfrac{d_2 · z}{...} }} $$ where $d_1 = \frac{10880}{13}$ , $d_2 = \frac{13810240}{247}$, $d_n = \frac14 \frac{(3n)(3n + 1)^2(3n + 2)^2(3n + 3)^2(3n + 4)} {(6n + 1)(6n + 7)} $ .
I have absolutely no idea how to even begin attacking this conjecture. It looks like something Ramanujan might have come up with. A search for “equiharmonic numbers” was futile.
Does anyone have any idea how this might be attacked or, even better, if this has been solved?