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Euler transform

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If two integer sequences and are related by[1]

or, in terms of generating functions and ,

then is said to be the Euler transform of .

Combinatorial interpretation

If is a nonnegative integer sequence with Euler transform then is the number of weakly increasing sequences of pairs such that and for all .

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


  • 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..