Note on derivation of order conditions for ARKN methods for perturbed oscillators

J.M. Franco derived the sufficient order conditions as well as the necessary and sufficient order conditions for his Adapted Runge–Kutta–Nyström methods (in short notation ARKN methods) based on the B-series theory [J.M. Franco, Runge–Kutta–Nyström methods adapted to the numerical integration of perturbed oscillators, Comput. Phys. Comm. 147 (2002) 770–787]. Unfortunately, some fundamental mistakes have been made in the derivation of order conditions in that paper. In view of importance of the algebraic order theory for ARKN methods, a new and correct derivation of the order conditions for the ARKN methods is presented in this short note.