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# Euler transform

## Definition

If two integer sequences ${\displaystyle \scriptstyle \{a_{1},a_{2},\dots \}\,}$ and ${\displaystyle \scriptstyle \{b_{1},b_{2},\ldots \}\,}$ are related by[1]

${\displaystyle 1+\sum _{n=1}^{\infty }b_{n}x^{n}=\prod _{i=1}^{\infty }{\frac {1}{(1-x^{i})^{a_{i}}}}\,}$

or, in terms of generating functions ${\displaystyle \scriptstyle A(x)\,}$ and ${\displaystyle \scriptstyle B(x)\,}$,

${\displaystyle 1+B(x)=\exp {\bigg [}\sum _{k=1}^{\infty }{\frac {A(x^{k})}{k}}{\bigg ]}\,}$

then ${\displaystyle \scriptstyle \{b_{n}\}\,}$ is said to be the Euler transform of ${\displaystyle \scriptstyle \{a_{n}\}\,}$.

## Notes

1. Sloane and Plouffe 1995, pp. 20-21.

## References

• Sloane, N. J. A. and Plouffe, S., The Encyclopedia of Integer Sequences, San Diego, CA, Academic Press, pp. 20-21, 1995.