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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 ${\displaystyle \scriptstyle \{a_{n}\}\,}$.

## Combinatorial interpretation

If ${\displaystyle \{a_{1},a_{2},\dots \}}$ is a nonnegative integer sequence with Euler transform ${\displaystyle \{b_{1},b_{2},\ldots \},}$ then ${\displaystyle b_{n}}$ is the number of weakly increasing sequences of pairs ${\displaystyle ((x_{1},y_{1}),...,(x_{k},y_{k}))}$ such that ${\displaystyle x_{1}+...+x_{k}=n}$ and ${\displaystyle 1<=y_{i}<=a(x_{i})}$ for all ${\displaystyle i}$.

For example, the third part of the Euler transform of {1,2,4,8,...} is A034691(3) = 7, corresponding to the following 7 sequences of pairs.

• ${\displaystyle ((1,1),(1,1),(1,1))}$
• ${\displaystyle ((1,1),(2,1))}$
• ${\displaystyle ((1,1),(2,2))}$
• ${\displaystyle ((3,1))}$
• ${\displaystyle ((3,2))}$
• ${\displaystyle ((3,3))}$
• ${\displaystyle ((3,4))}$

## 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.