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 \(d(n)\equiv (-1)^{n-1}\ (\text {mod }5)\) and other similar congruence relations.

中文翻译：

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Jacobi theta常数$$ \ theta _3 _ $θ3的泰勒系数

我们研究经典模块化形式的点（（x = 1 \））的Jacobi theta常数\（\ theta _3 \）的泰勒展开。这自然导致一个新的整数序列\（（（d（n））_ {n = 0} ^ \ infty = 1,1，-1,51,849，-26199，\ ldots \）作为泰勒系数出现在\（\ theta _3 \）的相关“居中”版本的扩展中。我们证明了关于数d（n）和猜想它们满足等价\（d（n）\ equiv（-1）^ {n-1} \（\ text {mod} 5）\）和其他类似结果的几个结果同余关系。