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New optimal, contractivity-preserving (CP), \(d\)-derivative, 2- and 3-step, predictor-corrector, Hermite-Birkhoff-Obrechkoff series methods, denoted by \(HBO(d,k,p)\), \(k=2,3\), with nonnegative coefficients are constructed for solving nonstiff first-order initial value problems \(y'=f(t,y)\), \(y(t_0)=y_0\). The upper bounds \(p_u\) of order \(p\) of \(HBO(d,k,p)\), \(k=2,3\) methods are approximately 1.4 and 1.6 times the number of derivatives \(d\), respectively. Their stability regions have generally a good shape and grow with decreasing \(p-d\). Two selected CP HBO methods: 9-derivative 2-step HBO of order 13, denoted by HBO(9,2,13), which has maximum order 13 based on the CP conditions, and 8-derivative 3-step HBO of order 14, denoted by HBO(8,3,14), compare well with Adams-Cowell of order 13 in PECE mode, denoted by AC(13), in solving standard N-body problems over an interval of 1000 periods on the basis of the relative error of energy as a function of the CPU time. They also compare well with AC(13) in solving standard N-body problems on the basis of the growth of relative error of energy and 10000 periods of integration. The coefficients of selected HBO methods are listed in the appendix.
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