  We study the Taylor expansion around the point $x=1$ of a classical modular
form, the Jacobi theta constant $\theta_3$. This leads naturally to a new
sequence $(d(n))_{n=0}^\infty=1,1,-1,51,849,-26199,\ldots$ of integers, which
arise as the Taylor coefficients in the expansion of a related "centered"
version of $\theta_3$. We prove several results about the numbers $d(n)$ and
conjecture that they satisfy the congruence $d(n)\equiv (-1)^{n-1}\
(\textrm{mod }5)$ and other similar congruence relations.
