The grand theta block coefficient formula is

B_{n}(φ)(ζ)
=∑_{u∈DP(n)}sgn(u)
∏_{i,j≥1}(1/u(i,j)!)
(G(φ)(ζ^{j})/j)^{u(i,j)}

Here φ:**Z**→**Z** is any even finitely supported function,
defining a corresponding product

P(φ)(τ,z)
=∏_{i≥1,r∈Z}
(1−q^{i}ζ^{r})^{φ(r)},
q=e^{2πiτ}
ζ=e^{2πiz}

and our formula gives the coefficients of the product's formal q-expansion

P(φ)(τ,z)
=∑_{n∈Z≥0}
B_{n}(φ)(ζ)q^{n}

In the formula, DP(n) is the set of double partitions of n,
and sgn(u) is the sign of the double partition u

DP(n)={u:**Z**_{≥1}x**Z**_{≥1}→**Z**_{≥0}:
n=∑_{i,j}u(i,j)ij},
sgn(u)=(−1)^{∑i,ju(i,j)}

and G is the germ of φ

G(φ)(ζ)=∑_{r∈Z}φ(r)ζ^{r}

Thus B_{n}(φ)(ζ)=b_{n}
(G(φ)(ζ),G(φ)(ζ^{2}),...,G(φ)(ζ^{n}))
where b_{n}∈(1/n!)**Z**[x_{1},x_{2},...,x_{n}]
is independent of φ