1,2

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 n-th power level, k-th 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 obtained 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 or r and z variables since the generated equations describe the same Euler/Pascal Cube series (and inform each other).

D. J. Pengelley, The bridge between the continuous and the discrete via original sources in Study the Masters: The Abel-Fauvel Conference [pdf], Kristiansand, 2002, (ed. Otto Bekken et al), National Center for Mathematics Education, University of Gothenburg, Sweden, in press.

C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube.

Eric Weisstein, Link to section of MathWorld: Worpitzky's Identity of 1883.

Eric Weisstein, Link to section of MathWorld: Eulerian Number.

Eric Weisstein, Link to section of MathWorld: Nexus number.

Eric Weisstein, Link to section of MathWorld: Finite Differences.

a(x) = Sum [Eulerian[n, i - 1]*Binomial[n + x - i + r, n + r], {i, 1, n}]; n = 5, r = -4, or a(x) = {-4 + 26*(-3 + x) + 66*(-2 + x) + 26*(-1 + x) + 2*x}; x>3, or a(x) = {-240 + 120*x}; x>3, or a(x) = {120*(-2 + x)}; x>3, or a(k) = Sum[(-1)^j*Binomial[n + 1 - z, j]*(k - j + 1)^n, {j, 0, k + 1}]; n = 5, z = 2, or a(k) = {120*(-1 + k)}; k>2

MagicNKZ=Sum[(-1)^j*Binomial[n+1-z, j]*(k-j+1)^n, {j, 0, k+1}]; Table[MagicNKZ, {n, 5, 5}, {z, 2, 2}, {k, 0, 34}] OR SeriesAtLevelR = Sum[Eulerian[n, i - 1]*Binomial[n + x - i + r, n + r], {i, 1, n}]; Table[SeriesAtLevelR, {n, 5, 5}, {r, -4, -4}, {x, 4, 35}]

Within the cube of related sequences with construction based upon MaginNKZ formula, with n downward, k rightward and z backward . . .:

Before: A101100, this_sequence, A101096, A101098, A022521, A000584, A000539, A101092, A101099

Above: A005408, A008458, A101103, this_sequence

Within the cube of related sequences with construction based upon SeriesAtLevelR formula, with n downward, x rightward, and r backward.

Above: A101104, this_sequence.

Cf. A047969.

Sequence in context: A042534 A042536 A042538 this_sequence A045823 A044360 A044741

Adjacent sequences: A101092 A101093 A101094 this_sequence A101096 A101097 A101098

easy,nonn,uned

Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004