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The ‘Borcherds products everywhere’ construction [Gritsenko et al., ‘Borcherds products everywhere’, J. Number Theory148 (2015), 164–195] creates paramodular Borcherds products from certain theta blocks. We prove that the 
$q$-order of every such Borcherds product lies in a sequence 
$\{C_{\unicode[STIX]{x1D708}}\}$, depending only on the 
$q$-order 
$\unicode[STIX]{x1D708}$ of the theta block. Similarly, the 
$q$-order of the leading Fourier–Jacobi coefficient of every such Borcherds product lies in a sequence 
$\{A_{\unicode[STIX]{x1D708}}\}$, and this is the sequence 
$\{a_{n}\}$ from work of Newman and Shanks in connection with a family of series for 
$\unicode[STIX]{x1D70B}$. Our proofs use a combinatorial formula giving the Fourier expansion of any theta block in terms of its germ.
