The grand theta block coefficient formula is

  Bn(φ)(ζ) =∑u∈DP(n)sgn(u) ∏i,j≥1(1/u(i,j)!) (G(φ)(ζj)/j)u(i,j)

Here φ:ZZ is any even finitely supported function, defining a corresponding product

  P(φ)(τ,z) =∏i≥1,r∈Z (1−qiζr)φ(r),   q=e2πiτ   ζ=e2πiz

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

  P(φ)(τ,z) =∑n∈Z≥0 Bn(φ)(ζ)qn

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≥1xZ≥1Z≥0: n=∑i,ju(i,j)ij},   sgn(u)=(−1)i,ju(i,j)

and G is the germ of φ

  G(φ)(ζ)=∑r∈Zφ(r)ζr

Thus Bn(φ)(ζ)=bn (G(φ)(ζ),G(φ)(ζ2),...,G(φ)(ζn)) where bn∈(1/n!)Z[x1,x2,...,xn] is independent of φ