The Taylor coefficients of the Jacobi theta constant ${\theta }_3$

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 $(d(n){)}_{n=0}^{\mathrm{\infty }}=1,1,-1,51,849,-26199,\dots$ 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 $d(n)\equiv (-1)^{n-1}
($\text{mod }$5)$ and other similar congruence relations.