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)
|