This work presents an algorithm for calculating high temperature series expansions (HTSE) of Heisenberg spin models with spin $S=1/2$ in the thermodynamic limit. This algorithm accounts for the presence of a magnetic field. The paper begins with a comprehensive introduction to HTSE and then focuses on identifying the bottlenecks that limit the computation of higher order coefficients. https://www.learningepistemology.com/product-category/supplements/