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Euler's Number Triangle

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The triangle of numbers A_(n,k) given by

A_(n,1)=A_(n,n)=1
(1)

and the recurrence relation

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

for k in [2,n], where A_(n,k) are shifted Eulerian numbers, i.e.,

<1; 0> <2; 0> <2; 1> <3; 0> <3; 1> <3; 2> <4; 0> <4; 1> <4; 2> <4; 3>
(3)
1 1 1 1 4 1 1 11 11 1 1 26 66 26 1 1 57 302 302 57 1 1 120 1191 2416 1191 120 1
(4)

(OEIS A008292). Note that the rows sum to the successive factorials 1=1!, 1+1=2!, 1+4+1=3!, 1+11+11+1=4!, ....

Binary plot of Euler's number triangle

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 is the generator for the first N rows of Euler's number triangle, when the ith term of the transform is first cleared of its denominator by multiplying through by (z-1)^(i+1). For example,

Z[{n^k}_(k=1)^3={z/((z-1)^2),(z+z^2)/((z-1)^3),(z+4z^2+z^3)/((z-1)^4)}.
(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

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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 Resource. https://mathworld.wolfram.com/EulersNumberTriangle.html

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