Summary: Computations with cumulants are becoming easier through the use of computer algebra but there remains a difficulty with the finiteness of the computations because all distributions except the normal have an infinite number of non-zero cumulants. One is led therefore to replacing finiteness of computations by ``finitely generated'' in the sense of recurrence relationships. In fact, it turns out that there is a natural definition in terms of the exponential model which is that the first and second derivative of the cumulant generating function, \(K\), lie on a polynomial variety. This generalises recent polynomial conditions on variance functions. This is satisfied by many examples and has applications to, for example, exact expressions for variance functions and saddle-point approximations.