Recently, one of the authors gave an algorithm for calculating the first nonzero Poincaré-Pontryagin function of a small polynomial perturbation of a polynomial Hamiltonian, under a generic hypothesis. We generalize this algorithm and show that any Poincaré-Pontryagin function of order l, denoted by Ml, can be written as a sum of an iterated integral of length at most l and of a combination of all previous Poincaré-Pontryagin functions, M1, M2, …, Ml-1, and their derivatives. This extends some results obtained recently and allows to identify the Bautin ideal with the ideal generated by iterated integrals.