  We extend some results recently obtained by Dan Romik about the Taylor
coefficients of the theta function $\theta_{3}\left(1\right)$ to the case
$\theta_{3}\left(q\right)$ of an arbitrary value of the elliptic modulus $k.$
These results are obtained by carefully studying the properties of the
cumulants associated to a $\theta_{3}$ (or discrete normal) distributed random
variable. This article also states some congruence conjectures about integers
sequences that generalize the one studied by D. Romik.
