This note proposes a new concept for the production of ultracold neutrons (UCNs) in a decelerating trap. UCNs are widely used in the physics of elementary particles and fundamental interactions, and can potentially be used in studies of condensed matter. However, most of these studies are limited by the available UCN densities and fluxes. One of the ways to increase them is to use peak fluxes in pulsed neutron sources, orders of magnitude larger than the mean values. Here, a concept of UCN sources is proposed, which allows to implement this idea. We propose to produce very cold neutrons (VCNs) in converters located in a neutron source, extract and slow them down to UCNs by a decelerating magnetic or material trap. As shown in this paper, for both pulsed and continuous neutron sources, this method could provide a high conversion efficiency of VCNs to UCNs with low losses of density in the phase space. More detailed calculations and the proposals for concrete technical designs are going to be developed in future publications.