Abelian ideals and cohomology of symplectic type

Let $\mathfrak{b}$ be a Borel subalgebra of the symplectic Lie algebra $\mathfrak{sp}(2n,\mathbb{C})$ and let $\mathfrak{n}$ be the corresponding maximal nilpotent subalgebra. We find a connection between the abelian ideals of $\mathfrak{b}$ and the cohomology of $\mathfrak{n}$ with trivial coefficients. Using this connection, we are able to enumerate the number of abelian ideals of $\mathfrak{b}$ with given dimension via the Poincaré polynomials of Weyl groups of types $A_{n-1}$ and $C_n$.