The exponential transform is the transformation of a sequence a_1, a_2, ... into a sequence b_1, b_2, ... according to the equation 1 + sum_(n = 1)^∞ (b_n x^n)/(n!) = exp( sum_(n = 1)^∞ (a_n x^n)/(n!)). The inverse ("logarithmic") transform is then given by sum_(n = 1)^∞ (a_n x^n)/(n!) = ln(1 + sum_(n = 1)^∞ (b_n x^n)/(n!)). The exponential transform relates the number a_n of labeled connected graphs on n nodes satisfying some property with the corresponding total number b_n (not necessarily connected) of labeled graphs on n nodes. In this application, the transform is called Riddell's formula for labeled graphs.
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