Multiparticle cumulants of azimuthal angle correlations have been compelling tools to probe the properties of the quark-gluon plasma (QGP) created in the ultrarelativistic heavy-ion collisions and the search for the QGP in small collision systems at BNL's Relativistic Heavy Ion Collider (RHIC) and at CERN's Large Hadron Collider (LHC). However, only very few of them are available and have been studied in theoretical calculations and experimental measurements, while the rest are generally very interesting, but their direct implementation was not feasible. In this paper, we present a generic recursive algorithm for multiparticle cumulants, which enables the calculation of arbitrary order single and mixed harmonic multiparticle cumulants. Among them, the new 10-, 12-, 14-, and 16-particle cumulants of a single harmonic, named c(n){10}, c(n){12}, c(n){14}, and c(n) {16}, and the corresponding v(n) coefficients, will be discussed for the first time. Our Monte Carlos studies show that these new multiparticle cumulants can be readily used along with updates to the generic framework of multiparticle correlations to a very high order. Finally, we propose a particular series of mixed harmonic multiparticle cumulants, which measures the general correlations between any moments of different flow coefficients. The predictions of these new observables are shown based on an initial-state model MC-Glauber, a toy Monte Carlo model, and the HIJING transport model for future comparisons between experimental data and theoretical model calculations. The study of these new multiparticle cumulants in heavy-ion collisions will significantly improve the understanding of the joint probability density function which involves both different harmonics of flow and also the symmetry planes. This will pave the way for more stringent constraints on the initial state and help to extract more precisely information on how the created hot and dense matter evolves. Meanwhile the efforts applied to small systems could be very helpful in the understanding of the origin of the observed collectivity at RHIC and the LHC.