The number of ad-nilpotent ideals of the Borel subalgebra of the classical Lie algebra of type B_n is determined using combinatorial arguments involving a generalization of Dyck-paths. We also solve a similar problem for the untwisted affine Lie algebra of type B_n, where we instead enumerate a certain class of ideals called basic ideals. This leads to an explicit formula for the number of basic ideals in B_n, which gives rise to a new integer sequence.