Recently, Romik determined in [8] the Taylor expansion of the Jacobi theta constant θ 3 , around the point x = 1. He discovered a new integer sequence, (d(n)) ∞ n=0 = 1, 1, −1, 51, 849, −26199, . . . , from which the Taylor coefficients are built, and conjectured that the numbers d(n) satisfy certain congruences. In this paper, we prove some of these conjectures, for example that d(n) ≡ (−1) n+1 (mod 5) for all n ≥ 1, and that for any prime p ≡ 3 (mod 4), d(n) vanishes modulo p for all large enough n.

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