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Seidel-Entringer-Arnold Triangle

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The Seidel-Entringer-Arnold triangle is the number triangle consisting of the Entringer numbers E_(n,k) arranged in "ox-plowing" order,

E_(00) E_(10)->E_(11) E_(22)<-E_(21)<-E_(20) E_(30)->E_(31)->E_(32)->E_(33) E_(44)<-E_(43)<-E_(42)<-E_(41)<-E_(40)

giving

1 0->1 1<-1<-0 0->1->2->2 5<-5<-4<-2<-0

(OEIS A008280).

Binary plot for the Seidel-Entringer-Arnold triangle

The plot above shows the binary representations for the first 255 (top figure) and 511 (bottom figure) terms of a flattened Seidel-Entringer-Arnold triangle.

See also

Bell Number, Boustrophedon Transform, Clark's Triangle, Entringer Number, Euler's Number Triangle, Leibniz Harmonic Triangle, Losanitsch's Triangle, Number Triangle, Pascal's Triangle

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References

Arnold, V. I. "Bernoulli-Euler Updown Numbers Associated with Function Singularities, Their Combinatorics, and Arithmetics." Duke Math. J. 63, 537-555, 1991.Arnold, V. I. "Snake Calculus and Combinatorics of Bernoulli, Euler, and Springer Numbers for Coxeter Groups." Russian Math. Surveys 47, 3-45, 1992.Conway, J. H. and Guy, R. K. In The Book of Numbers. New York: Springer-Verlag, 1996.Dumont, D. "Further Triangles of Seidel-Arnold Type and Continued Fractions Related to Euler and Springer Numbers." Adv. Appl. Math. 16, 275-296, 1995.Entringer, R. C. "A Combinatorial Interpretation of the Euler and Bernoulli Numbers." Nieuw Arch. Wisk. 14, 241-246, 1966.Millar, J.; Sloane, N. J. A.; and Young, N. E. "A New Operation on Sequences: The Boustrophedon Transform." J. Combin. Th. Ser. A 76, 44-54, 1996.Seidel, I. "Über eine einfache Entstehungsweise der Bernoullischen Zahlen und einiger verwandten Reihen." Sitzungsber. Münch. Akad. 4, 157-187, 1877.Sloane, N. J. A. Sequence A008280 in "The On-Line Encyclopedia of Integer Sequences."

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Seidel-Entringer-Arnold Triangle

Cite this as:

Weisstein, Eric W. "Seidel-Entringer-Arnold Triangle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Seidel-Entringer-ArnoldTriangle.html

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