"In his correspondence of 1636, Pierre de Fermat called the problem of finding formulas for sums of powers "what is perhaps the most beautiful problem of all arithmetic", and claimed a recursive solution using figurate numbers, which could then be applied to integrate the "higher parabolas" x^k (Boyer, 1943; Katz, 1998, p. 481ff)."     Pengelley, p. 4

An example of 6⁴ = 1296, defined across figurate series

Figure 1.  Definition of the 6th value of the power of 4 according to the Euler/Pascal "SeriesAtLevelR" formula (or Worpitzky's identity):

where n-PT refers to the nth figurate series of Pascal's Triangle.  [By applying the same "weighting" procedure to the parallel 6-PT figurate numbers--rather than the 5-PT numbers--the defined amount would be the summation of the 1st through 6th (integer) values for x⁴.  Similarly, applying the procedure to the parallel 5-PT figurate numbers defines the 6th shell or nexus number that is one of the 1st through 6th set that add up to an x⁴ value. The procedure applies across figurate series from the ones onward.  It generates recursively accumulating series of values, all dependent upon the single set of "weights".]

Figure 2.  This is a logical parsing of the 6⁴ value across several Pascal Triangle figurate series.  The parsing is based on the definition (see figure 1) which "starts" here in the rightmost columns.  The logic:  the 1st through xth of figurate series define the xth of next higher figurate series, so "weighting" can be translated.

Amazing number of figurate number renditions of a power-value, isn’t it?  Do you see how the Euler Triangle row numbers that are the multipliers defining x⁴ values "accumulate" as they are applied to previous series?  The x^th of a PT series can also be called the 1^st through x^th of previous figurate series, so, if the x^th of (n+1)-PT is treated by a multiplier in a definition, then the multiplier similarly affects the 1^st through x^th of the previous, n-PT series. 

Figure 3.  Graph of the weighting of n-PT figurate values that generate shells or nexus number values which sum up to define a value that is k⁴.  Each row shows a specific shell value; kth value at top through the 1st "1" at bottom.  Each diagonal, running from top-right to bottom-left, depicts a single 'k' or 'k minus something' -th of the n-PT figurate numbers.  The pattern holds across powers diagramming n columns for the power of n and k rows for the kⁿ value.

Using values of the 4th figurate series of Pascal's Triangle, weighted to sum up to k⁴:

1 kth value +

12 (k - 1)th values +

23 (k - 2)th values +

24 (k - 3)th values +

24 (k - 4)th values +

etc. until

24 (k - (k-1))th value (which is the first "1")

Figure 4.  Chart of accumulation that occurs from Euler's Triangle, row 4 (that defines series of values regarding the power of 4).  See the diagonals of figure 3 which demonstrate the mechanism of the accumulation from an initial 1, 11, 11, and 1.

The amount of numbers to generate rightward in figure 4 would depend on the x^th being examined.  The appropriate level of accumulation depends on how many Pascal Triangle levels lower that an initial PT level (plied with a Euler row weighting) would be the desired figurate definition. In other words, each accumulation parses a definition into the next lower-level figurate-number definition.

Notice that an x^n value at the 1-PT level is shell/nexus number multiplying terms (of ones) that sum to x^n.  Notice that the smallest valued term is the xth term and values grow larger back through to the largest that is the 1st term.  Simultaneously, application of Euler's Triangle row n to n-PT values (see method of figure 2 applied to the example within figure 3) the same series is generated, however the smallest term is the first and the largest term is in the xth position.

The space in figure 4 between the 5^th and 6^th series shows where the usage of accumulating Euler row multipliers broke off in figure 2's example of 6⁴.  Whenever Euler Triangle row n is applied to the (n+1)^th series of the Pascal Triangle, not only will xⁿ values result across Pascal Triangle series as shown in figure 2, but 1-PT ones also will always finally be multiplied by the series that is the shell/nexus number series to that power of n.  This is a confirmation of shells/nexus number roles even within Worpitzky’s identity.  This also points out that, in a larger sense, the identity is a but a single case of a whole, coherent Euler/Pascal cube relationship.

General Definitions of Powers (then shells/nexus numbers) Extending Across Figurate Series

Statement 0 gives the layout for the power of 0.

Table 1 gives the layout for the power of 1.

Table 2 gives the layout for the power of 2.

Table 3 gives the layout for the power of 3.

Table 4 gives the general case.

Figures 1 - 4 give The Example of 6^4 and explanation of the pattern.

Statement 0.

Power of 0:  (0+1)th figurate series of the Pascal Triangle that is 1's is not weighted by a row of Euler's Triangle.  There are no shells.  Ones are ones.

Table 1.

*(A moot weighting affirming that integers exist and they sum from 1's.)

Table 2.

*Interestingly, this Euler Triangle "weighting" of the integer series can be construed to be represented by the same equation as that for the summed shell series in the box below it.  In no later case of the powers is the definition that is the nth-figurate series' version of a power's value represent-able by the summation of an equation.  Instead, these values are determined in the higher powers by the summation of an algorithmic application of "Euler series" and are ultimately not elements describable by an equation. 

The rule for the asterisked series is:  the summation of '1 through (x-1)' in the equation '2x' plus the xth integer that is 'x' (--But x could be considered to be half of 2x . . .)  So you may as well have said the summation was of '1 through x' in the equation '2x' and 1 was subtracted at each summation for a total of x subtractions of '1' (dispensing of an extra x amount).

Table 3.

Table 4.

The General Pattern of Figurate Definitions of Power Values and Shell/Nexus Number Values of a Power Across Various Figurate Series — which is the pattern for all applications of "Seed number weightings" including application to higher PT levels thus defining summations ("partial sums") of powers and summations of summations of powers . . . (partial sums of powers and higher are not depicted below).

Assuming an alphabet representation of series where:

Series named A through Z represent recursively related series where A is given and B through Z give n amount of new series that are based on summations of previous series. (Z is a member of an alphabet that may have missing or extra letters than the normal alphabet, but Z is still the last letter.)

Each of the 1st through xth of the A ... Z-series' values align with each of the xth through the 1st of an appropriate Pascal Triangle figurate series (as graphed below by rows) for a one-to-one cross application of values of the two series as multipliers of each other.  These multiplications create terms to sum that then define the xth value of the series (named at the top of the column).

Table 4

Depiction of the general figurate and Euler number relationships for series regarding specific powers:

Assuming ones are ones, the definitions of all series regarding powers are logically traced to the assumption that ones are ones and rows of Euler's Triangle meaningfully and consistently are "seeders" of series regarding powers.  Simultaneous dual, last-to-1st then 1st-to-last, definitions of value-generating series regarding a power level's values is the powerful basis of logical continuity that is simply applications of "seed" weights to any or all PT levels. 

The bottom row of the chart gives series described in conjunction with 1st-figurate level of 1's as multipliers.  These series are fundamental and (moving from right to left) simply recursive accumulations from series to next series beginning with "seed" values (A) that are one of the rows from Euler's Triangle. 

(Incidentally, each Euler Triangle row describes permutations [possible orderings] of n things as sub-divided into categories of increasing or decreasing orderliness).

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