Euler transform

Definition

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

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

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

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

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

Combinatorial interpretation

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

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.

• $((1,1),(1,1),(1,1))$
• $((1,1),(2,1))$
• $((1,1),(2,2))$
• $((3,1))$
• $((3,2))$
• $((3,3))$
• $((3,4))$

References

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

External links

• Eric W. Weisstein, Euler Transform, from MathWorld — A Wolfram Web Resource..