Boustrophedon Transform

The boustrophedon (``ox-plowing'') transform b of a sequence a is given by

$$b_n = \sum_{k=0}^n {n\choose k} a_kE_{n-k}$$ (1)

$$a_n = \sum_{k=0}^n (-1)^{n-k}{n\choose k}b_k E_{n-k}$$ (2)

for $n\geq 0$, where $E_n$ is a Secant Number or Tangent Number defined by

$$\sum_{n=0}^\infty E_n{x^n\over n!}=\sec x+\tan x.$$ (3)

The exponential generating functions of a and b are related by

$${\mathcal B}(x)=(\sec x+\tan x){\mathcal A}(x),$$ (4)

where the exponential generating function is defined by

$${\mathcal A}(x)=\sum_{n=0}^\infty A_n{x^n\over n!}.$$ (5)

See also Alternating Permutation, Entringer Number, Secant Number, Seidel-Entringer-Arnold Triangle, Tangent Number

References

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.