uses a recurrence relation. Define the numbers (with k ≥ n ≥ 0) by . Then the transformed sequence is defined by (for and greater indices). Per this definition, note the following definitions for values outside the restrictions (from the relationship above) on pairs: Special Cases In the case a0 = 1, an = 0 (n > 0), the resulting triangle is called the Seidel–Entringer–Arnold Triangle and the numbers are called Entringer numbers . In this case the numbers in the transformed sequence bn are called the Euler up/down numbers. This is sequence A000111 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 () to output values () can be defined for different algebras ("numeric domains"). Euclidean (Real) values In the Euclidean () Algebra for Real ()-valued scalars, the boustrophedon transformed Real-value is related to the input value, , as: , with the reverse relationship (input from output) defined as: , 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 (an) is defined by The exponential generating function of the boustrophedon transform (bn) is related to that of the original sequence (an) by The exponential generating function of the unit sequence is 1, so that of the up/down numbers is sec x + tan x. References Integer sequences Triangles of numbers Permutations Transforms