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 {n^k}_(k=1)^N" src="https://mathworld.wolfram.com/images/equations/EulersNumberTriangle/Inline8.gif" style="height:22px; width:42px" /> 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,

{n^k}_(k=1)^3={z/((z-1)^2),(z+z^2)/((z-1)^3),(z+4z^2+z^3)/((z-1)^4)}. " src="https://mathworld.wolfram.com/images/equations/EulersNumberTriangle/NumberedEquation5.gif" style="height:43px; width:282px" />

(5)

REFERENCES:

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