Symbolic derivation of order conditions for hybrid Numerov-type methods solving y″=f(x,y)y″=f(x,y)

Numerov-type ODE solvers are widely used for the numerical treatment of second-order initial value problems. In this work we present a powerful and efficient symbolic code in MATHEMATICA for the derivation of their order conditions and principal truncation error terms. The relative tree theory for such order conditions is presented along with the elements of combinatorial mathematics, partitions of integer numbers and computer algebra which are the basis of the implementation of the symbolic code.