The coagulation equation with kernel Ko = A + B(i + j) + Cij and arbitrary initial conditions is studied analytically and a simple expression for the solution is found. For monodisperse initial conditions, we recover the known size distribution expressed in terms of a degeneracy factor N_k, which is determined by a recursion relation. For polydisperse initial conditions, a similar solution form is found, which includes a degeneracy factor N_kl, also determined by a recursion relation. The physical meaning of N_kl and the recursion relation is given. A method to get explicit expressions for N_k and N_kl is illustrated. Finally, the pre-gel solution is given explicitly and a general method to get the post-gel solution is proposed.