Series Reversion

Author

Eric W. Weisstein
April 15, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Calculus/SeriesReversion.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/SeriesReversion.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Formula

(Subscript[a, n])==(1/(n (Subscript[a, 1]^n)))(∑_(s,t,u,…))(((((-1))^(s+t+u+…)) n (n+1) ⋯ (n+s+t+u+…-1) )/(s! t! u! ⋯))(((((Subscript[a, 2])/(Subscript[a, 1]))))^s) (((((Subscript[a, 3])/(Subscript[a, 1]))))^t) ⋯

where

s+2 t+3 u+…==n-1

Terms

With[{n=7},
CoefficientList[InverseSeries[SeriesData[x,0,Array[a,n],1,n+1,1]],x]
]//ColumnForm