Euler's Triangle

The triangle of numbers $A_{n,k}$ given by

$$A_{n,1}=A_{n,n}=1$$

and the Recurrence Relation

$$A_{n+1,k}=kA_{n,k}+(n+2-k)A_{n,k-1}$$

for $k\in [2,n]$, where $A_{n,k}$ are Eulerian Numbers.

$1$
$1\quad 1$
$1\quad 4\quad 1$
$1\quad 11\quad 11\quad 1$
$1\quad 26\quad 66\quad 26\quad 1$
$1\quad 57\quad 302\quad 302\quad 57\quad 1$

The numbers 1, 1, 1, 1, 4, 1, 1, 11, 11, 1, ... are Sloane's A008292. Amazingly, the Z-Transform of $t^n$

$${(z-1)^n\over T^n z}Z[t^n]={(1-z)^n\over T^n z} \lim_{x\to 0}{\partial^n\over\partial x^n}\left({z\over z-e^{-xT}}\right)$$

are generators for Euler's triangle.

See also Clark's Triangle, Eulerian Number, Leibniz Harmonic Triangle, Number Triangle, Pascal's Triangle, Seidel-Entringer-Arnold Triangle, Z-Transform

References

Sloane, N. J. A. Sequence A008292 in ``The On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.