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The front of the cube is Euler's Triangle. The top of the cube is Pascal's Triangle.
Two versions of the cube each have a formula that describes any of its series in any one of the three directions. The two formulas are dubbed "MagicNKZ" and "SALR" in these pages.
“SeriesAtLevelR” and “MagicNKZ” produce logically reciprocating formulas when solving for either x and k and/or r and z variables since the differently generated equations describe the same Euler/Pascal Cube series (and inform each other).
Series in yellow are xn values. Series that run left to right recursively sum to values of the next series after.
MagicNKZ formula:
The (Worpitzky/Euler/Pascal Cube) “MagicNKZ” algorithm is: Sum [(-1)^j*Binomial[n + 1 - z, j]*(k - j + 1)^n, {j, 0, k+1}] Offset: (0,1,0) for (k, n, z) SEE EXAMPLE PAGE
In order to generate sequences that are Euler Triangle rows (see A008292) or accumulations from Euler Triangle rows, values of Euler/Pascal cube series are defined with the variables of nth power level, kth order of occurrence and zth accumulation level. The generating algorithm is based upon binomial definitions for Euler Triangle values. Rows of Euler’s Triangle are given when z = 0. The number of summations from an initial Euler Triangle row is “enumerated” by the value that is z.
SALR formula:
The (Worpitzky/Euler/Pascal Cube) “SeriesAtLevelR” algorithm is: Sum [Eulerian[n, i - 1]*Binomial[n + x - i + r, n + r], {i , 1, n}] Offset: (1,1, 0-->relative to the powers) for (x, n, r) SEE EXAMPLE PAGE
Due to figurate number content within recursively accumulating series that exist for each power level, Worpitzky’s identity of 1883—which is based upon figurate numbers—not only will define values that are power values, but also will logically apply to sequences that are shells (of shells) [i.e., nexus numbers or difference sequences of powers] as well as summations (of summations) of power series. The “SeriesAtLevelR” algorithm is Worpitzky’s ID at r = 0 and defines nexus number/shell values when r < 0 and summations (of summations . . .) of powers when r > 0.
If r is positive, the series are summations or summations of summations of power series. If r is negative, the series are shell (nexus number) series or shell of shell series. x is the xth in series. n is the power level.
SUMMARY OF THE EULER/PASCAL CUBE(s):
1) The (Worpitzky/Euler/Pascal Cube) “SeriesAtLevelR” algorithm is: Sum [Eulerian[n, i - 1]*Binomial[n + x - i + r, n + r], {i , 1, n}] Offset: (1,1, 0-->relative to the powers) for (x, n, r)
Due to figurate number content within recursively accumulating series that exist for each power level, Worpitzky’s identity of 1883—which is based upon figurate numbers—not only will define values that are power values, but also will logically apply to sequences that are shells (of shells) [i.e., nexus numbers or difference sequences of powers] as well as summations (of summations) of power series. The “SeriesAtLevelR” algorithm is Worpitzky’s ID at r = 0 and defines nexus number/shell values when r < 0 and summations (of summations . . .) of powers when r > 0.
2) The (Worpitzky/Euler/Pascal Cube) “MagicNKZ” algorithm is: Sum [(-1)^j*Binomial[n + 1 - z, j]*(k - j + 1)^n, {j, 0, k+1}] Offset: (0,1,0) for (k, n, z)
In order to generate sequences that are Euler Triangle rows (see A008292) or accumulations from Euler Triangle rows, values of Euler/Pascal cube series are defined with the variables of nth power level, kth order of occurrence and zth accumulation level. The generating algorithm is based upon binomial definitions for Euler Triangle values. Rows of Euler’s Triangle are given when z = 0. The number of summations from an initial Euler Triangle row is “enumerated” by the value that is z.
“SeriesAtLevelR” and “MagicNKZ” produce logically reciprocating formulas when solving for either x and k and/or r and z variables since the differently generated equations describe the same Euler/Pascal Cube series (and inform each other).
Formulas:A Mathematica Notebook of Formulas
Two-Variable Equations
Single-Variable Equations in concise equation layouts with depictions of series and with links to Sloane's On-line Encyclopedia of Integer Sequences
Parallel (same) sequences:
Unrelated (different) sequences:
Matrices of numbers/values
More Single-Variable equations SeriesAtLevelR (large file) MagicNKZ (large file) Yet more single variable equations SeriesAtLevelR (large file) MagicNKZ (large file) |
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