A Formal Solution of Σ∞i=1AieBix = X
George N. Raney · Canadian Journal of Mathematics · 1964
We consider a combinatorial enumeration problem involving certain collections of labelled rooted trees having coloured nodes and edges. Notations and definitions are introduced in §§2 and 3, and the problem is described in §4. We give recursion formulas for its solution in §5. Then, by using a modification of a method of Priifer, we obtain a direct solution in terms of multinomial coefficients and power products in §§6 and 7. These results are combined in §8. Working in a formal power series algebra we find a formal solution of the equation which expresses the unknown X as a multiple power series in the Ai and the Bi.