The Taylor coefficients of the Jacobi theta constant θ _3

We study the Taylor expansion around the point x=1 of a classical modular form, the Jacobi theta constant θ _3 . This leads naturally to a new sequence (d(n))_n=0^∞ =1,1,-1,51,849,-26199,… of integers, which arise as the Taylor coefficients in the expansion of a related “centered” version of θ _3 . We prove several results about the numbers d ( n ) and conjecture that they satisfy the congruence d(n)≡ (-1)^n-1 (mod 5) and other similar congruence relations.