  Recently, Romik determined in [9] the Taylor expansion of the Jacobi theta
constant \theta_3, around the point x = 1. He discovered a new integer
sequence, (d(n))_0^\infty=1, 1, -1, 51, 849, -26199, \dots, from which the
Taylor coefficients are built, and conjectured that the numbers d(n) satisfy
certain congruences modulo various primes. In this paper, we prove some of
these conjectures, for example that d(n)\equiv (-1)^{n+1}(mod 5) for all n\geq
1,and that for any prime p\equiv 3 (mod 4), d(n) vanishes modulo p for all
large enough n.
