  Based on an identity of Jacobi, we prove a simple formula that computes the
pushforward of analytic functions of the exceptional divisor of a blowup of a
projective variety along a smooth complete intersection with normal crossing.
We apply this pushforward formula to derive generating functions for Euler
characteristics of crepant resolutions of singular Weierstrass models given by
Tate's algorithm. Since these Euler characteristics depend only on the sequence
of blowups and not on the Kodaira fiber itself, nor the associated group,
several distinct Tate models have the same Euler characteristic. In the case of
elliptic Calabi-Yau threefolds, we also compute the Hodge numbers. For
elliptically fibered Calabi-Yau fourfolds, our results also prove a conjecture
of Blumenhagen-Grimm-Jurke-Weigand based on F-theory/heterotic string duality.
