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Exponential Transform

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)^infty(b_nx^n)/(n!)=exp(sum_(n=1)^infty(a_nx^n)/(n!)).

The inverse ("logarithmic") transform is then given by

sum_(n=1)^infty(a_nx^n)/(n!)=ln(1+sum_(n=1)^infty(b_nx^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.

See also

Binomial Transform, Euler Transform, Logarithmic Transform, Möbius Transform, Riddell's Formula, Stirling Transform

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References

Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, pp. 19-20, 1995.

Referenced on Wolfram|Alpha

Exponential Transform

Cite this as:

Weisstein, Eric W. "Exponential Transform." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ExponentialTransform.html

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