Boustrophedon transform

In mathematics, the boustrophedon transform is a procedure which maps one sequence to another. The transformed sequence is computed by an "addition" operation, implemented as if filling a triangular array in a boustrophedon (zigzag- or serpentine-like) manner—as opposed to a "Raster Scan" sawtooth-like manner.
Definition.
The boustrophedon transform is a numerical, sequence-generating transformation, which is determined by an "addition" operation.
Generally speaking, given a sequence: formula_1, the boustrophedon transform yields another sequence: formula_2, where formula_3 is likely defined equivalent to formula_4. The entirety of the transformation itself can be visualized (or imagined) as being constructed by filling-out the triangle as shown in Figure 1.
Boustrophedon Triangle.
To fill-out the numerical Isosceles triangle (Figure 1), you start with the input sequence, formula_1, and place one value (from the input sequence) per row, using the boustrophedon scan (zigzag- or serpentine-like) approach.
The top vertex of the triangle will be the input value formula_4, equivalent to output value formula_3, and we number this top row as row 0.
The subsequent rows (going down to the base of the triangle) are numbered consecutively (from 0) as integers—let formula_8 denote the number of the row currently being filled. These rows are constructed according to the row number (formula_8) as follows:
Refer to the arrows in Figure 1 for a visual representation of these "addition" operations.
For a given, finite input-sequence: formula_26, of formula_27 values, there will be exactly formula_27 rows in the triangle, such that formula_8 is an integer in the range: formula_30 (exclusive). In other words, the last row is formula_31.
Recurrence relation.
A more formal definition uses a recurrence relation. Define the numbers formula_32 (with "k" ≥ "n" ≥ 0) by
Then the transformed sequence is defined by formula_38 (for formula_39 and greater indices).
Per this definition, note the following definitions for values outside the restrictions (from the relationship above) on formula_40 pairs:
formula_41
Special Cases.
In the case "a"0 = 1, "a""n" = 0 ("n" > 0), the resulting triangle is called the Seidel–Entringer–Arnold Triangle and the numbers formula_32 are called Entringer numbers .
In this case the numbers in the transformed sequence "b""n" are called the Euler up/down numbers. This is sequence on the On-Line Encyclopedia of Integer Sequences. These enumerate the number of alternating permutations on "n" letters and are related to the Euler numbers and the Bernoulli numbers.
Algebraic Definition(s).
Building from the geometric design of the boustrophedon transform, algebraic definitions of the relationship from input values (formula_43) to output values (formula_44) can be defined for different algebras ("numeric domains").
Euclidean (Real) values.
In the Euclidean (formula_45) Algebra for Real (formula_46)-valued scalars, the boustrophedon transformed Real-value is related to the input value, , as:
formula_47,
with the reverse relationship (input from output) defined as:
formula_48,
where is the sequence of "up/down" numbers—also known as secant or tangent numbers.
The exponential generating function.
The exponential generating function of a sequence ("a""n") is defined by
The exponential generating function of the boustrophedon transform ("b""n") is related to that of the original sequence ("a""n") by
The exponential generating function of the unit sequence is 1, so that of the up/down numbers is sec "x" + tan "x".