"SeriesAtLevelR" FORMULAS WITHIN THE EULER/PASCAL CUBE: Two Variables at a Time
(in formats from the Mathematica program, from Wolfram Research, Inc.)
Figure 1. The Euler/Pascal Cube of accumulating series in the "SeriesAtLevelR" layout. Pascal's Triangle is the top layer extending backward.
"SeriesAtLevelR":
<<DiscreteMath`Combinatorica`
This is the "three-dimensional" version of the formula for Euler's Triangle: for an xth value of the power level n at r level of accumulation. When r = 0, a power series (in yellow) results, when r < 0, a shell series results, and when r > 0, a summation of powers series results.
Figure 2.
Planes from top to bottom of the cube of SeriesAtLevelR formulation of the Euler/Pascal cube give sets of series that are each specific to a power level of n. Each layer of the cube as depicted in figure 2 may be considered at a power's level and is presented in table 1 as one equation per power. Each equation solves for shells (of shells) series, power series, and summations (of summations) of powers series—depending on r. Notice that coefficients to the binomial variables of the per-power-equations are Euler's Triangle.
Table 1. Each formula solves for one of the layers occurring from top to bottom within the SeriesAtLevelR cube depicted in figure 2.
Figure 3.
Sets of (recognizable) formulas arranged by sheets of a Euler/Pascal cube layout according to the SeriesAtLevelR formulation. Each 'wall' of formulas is at one accumulation level (r) of either shell to powers, powers, or summation of powers. Given series of values range across power levels but are at one level of r per wall.
Table 2. Each formula solves for one of the walls from front to back within the SeriesAtLevelR cube depicted in figure 3.
Figure 4.
Table 3. Each formula solves for one of the upright sheets occurring from left to right within the SeriesAtLevelR cube depicted in figure 4.
