We derive closed recursion equations for the symmetric polynomials occurring in the form factors of Dn(1) affine Toda field theories. These equations follow from kinematical and bound state residue equations for the full form factor. We also discuss the equations arising from second- and third-order forward channel poles of the S-matrix. The highly symmetric case of D4(1) form factors is treated in detail. We calculate explicitly cases with a few particles involved.