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# Euler's Number Triangle

The triangle of numbers given by

 (1)

and the recurrence relation

 (2)

for , where are shifted Eulerian numbers, i.e.,

 (3)
 (4)

(OEIS A008292). Note that the rows sum to the successive factorials , , , , ....

The plot above shows the binary representations for the first 255 (top figure) and 511 (bottom figure) terms of a flattened Euler's number triangle.

Amazingly, the Z-transform of is the generator for the first rows of Euler's number triangle, when the th term of the transform is first cleared of its denominator by multiplying through by . For example,

 (5)

## See also

Clark's Triangle, Eulerian Number, Leibniz Harmonic Triangle, Losanitsch's Triangle, Number Triangle, Pascal's Triangle, Second-Order Eulerian Triangle, Seidel-Entringer-Arnold Triangle, Spherical Triangle, Z-Transform

## Explore with Wolfram|Alpha

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## References

Sloane, N. J. A. Sequence A008292 in "The On-Line Encyclopedia of Integer Sequences."

## Referenced on Wolfram|Alpha

Euler's Number Triangle

## Cite this as:

Weisstein, Eric W. "Euler's Number Triangle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EulersNumberTriangle.html