NOTICING NUMBERS:

ELEMENTARY POWER FORMULAS FOR POSITIVE INTEGERS

Add two neighboring triangular numbers for a square.  [An x^th triangular number equals x(x+1)/2.  The values appear in series in Pascal’s Triangle in the third figurate series.]   It is not as frequently recognized that two neighboring tetrahedral numbers equal a summation of squares, but they do.  [An x^th tetrahedral number is x(x+1)(x+2)/6—the fourth series of Pascal’s Triangle.]   In fact the pattern continues.   A sense of “raised” values—in this case, the power of 2—directly relate to figurate content of series in Pascal’s Triangle. 

Figure 1.  Pascal’s Triangle—either columns or rows contain figurate numbers in series.  The “top” of the triangle is the upper left corner.

Figurate numbers were an important topic of discussion for Pierre de Fermat and Blaise Pascal in their 17^th-century correspondence.  Pascal’s Triangle is recursive.  An x^th value of any series is the sum of the 1^st through x^th of the previous series.  Pascal promised to publish a method for attaining the sums of powers.  However, it was Bernoulli, after Pascal, who perfected a rather complicated technique for formulating sums of powers when Pascal did not quite clarify the pattern. 

WHAT IS THE PATTERN FOR POWER SERIES?

Two neighboring triangular numbers are squares; two neighboring tetrahedral numbers are a summation-of-squares; two neighboring numbers of the 5^th-figurate series of Pascal’s Triangle equal a summation-of-summations-of-squares; etc.  (Two neighboring integer, 2^nd-series numbers equal a shell/nexus number of squares—one number of the series that would accumulate to a square.)

How is this possible?

The manner in which Pascal Triangle’s figurate numbers self-referentially sum to each next level of figurate numbers is also the manner in which series accumulate to powers.  Working backward, any power of n accumulates initially from constants that are recognizable as row n of Euler’s Triangle.  Eulerian “seed numbers” generate powers’ series!

Table 1.  Examples of Accumulation from Euler “Seed Numbers”:  Powers of 3 through 7.  The table is attained by finite differences—working backward from power series to shell/nexus number series to all previous series.  Each power’s differences culminate at a top line that is a row of Euler’s Triangle followed by zeros.

ACCUMULATION GENERATES FIGURATE CONTENT IN POWER SERIES

Not only will two neighboring triangular numbers then two neighboring tetrahedral numbers then two neighbors, on up the Pascal Triangle figurate series, be values of summations of summations, but sets of three tetrahedral numbers equal a cube while sets of three 5^th-polytope-numbers (Pascal Triangle figurate numbers are also called polytope-numbers) equal a summation of cubes, and sets of three 6^th-polytope-numbers equal the summation of sums of cubes, etc.

Sets of four 5^th-polytope-numbers will equal an x⁴ value while sets of four 6^th-polytope-numbers will equal a summation of x⁴ values.

Sets of five 6^th-polytope-numbers will equal an x⁵ value while sets of five 7^th-polytope-numbers will equal a summation of x⁵ values.

Et cetera, ad infinitum . . .

A PRECISE DECRYPTION OF FIGURATE CONTENT IS EMBEDDED IN EULER’S TRIANGLE

Euler’s Triangle serves as a decryption key to figurate number content within power series values.  If Euler row n weights, by multiplication, the values of (n+1)^th-polytope-numbers [or (n+1)^th figurate series of Pascal’s Triangle], it describes a value that is xⁿ.  If Euler row n weights (n+2)^th polytope-numbers, a summation of the 1^st through x^th of xⁿ is described.  If Euler row n weights (n+3)^th polytope-numbers, a summation of the 1^st through x^th of summations of xⁿ is described.  If Euler row n weights n^th polytope-numbers, a shell or nexus number series value for the power of n is described.

Thus all powers, sums of powers, as well as higher summations and lower nexus/shell numbers are construable figurate values.  Series organize from both Pascal Triangle figurate series and Euler Triangle rows to form a cube. 

Figure 2.  The Euler/Pascal Cube of accumulating series.  Pascal's Triangle would be the top layer extending backward in this graphic.

THE FORMULAS (in format of the Mathematica program, from Wolfram Research, Inc.)

"Two Dimensional" formula for the Euler Triangle:  (k-1)^th of row n

The "three-dimensional" version of the formula for Euler's Triangle:  for a (k-1)^th value of power level n at z level of accumulation from the Euler/Pascal into the Euler/Pascal Cube

"Two dimensional" formula for Worpitzky's identity of 1883 which produces xⁿ (using a Mathematica add-in):

<<DiscreteMath`Combinatorica`

The "three dimensional" formula for an x^th value in the power level n within the Euler/Pascal Cube.  When r = 0 it is a power series (in yellow), when r < 0 it is a shell series, and when r > 0 it is a summation of powers series:

Combinatorial Meaning Part 1 of 2

PASCAL’S TRIANGLE SERIES REPRESENT PERMUTATIONS

Each level of Pascal Triangle figurate series (also called polytope-numbers) can be said to represent a certain sized set as a category of permutations or arrangements of things.  The total number of items from which to make permutations increases by one item per row from the pinnacle of Pascal’s Triangle.  Order does not matter. 
 

Figure 3.  Pascal’s Triangle:  Sets of permutations when including one new variable-item per row (Rows appear diagonally)

Thus power series values are “about” set-size categories if they are “about” Pascal Triangle figurate numbers.  Specific figurate series represent particular sized “packets”.  The x^th case of a power reflects the impact that the total numbers of items from which packets are contrived contributes to populating packets.  A figurate value is the number of new permutations possible in that figurate set size, considering the total items at the x^th level.  Definitions of xⁿ are based on the (n+1)^th figurate numbers of Pascal’s Triangle and fundamentally refer to new permutations at the level x-total-items of set-size-(n+1) [as well as the previous (n-1) levels of new permutations, i.e., several previous figurate numbers of the series.  This is explained next and formulas for the layers are given in Figure 4.]

Combinatorial Meaning Part 2 of 2

THE IMPORTANCE OF EULER’S TRIANGLE

Euler’s Triangle represents all possible either ascending or descending orderliness, given n items, when grouped by “runs” of ascents or descents. 

Figure 3.  Euler’s Triangle:  The number of ascents per n items, grouped by runs

The application of Euler Triangle numbers (by row) to Pascal Triangle figurate series is the source of the pattern that two figurate numbers define squares, three define cubes, four define the fourth power, etc.  Euler number application patterns lead to n amount of terms defining shells or sums of powers or, in fact, any value in any of a series summing to or from or at the power of n.  (The powers' pattern of sieved accumulation graphs another marvelous pattern that visually reinforces a concept that every xⁿ series is about n, as well as (n-1), (n-2) . . . 1.) 

Figurate numbers describe the correct amount of sets of each of the Euler row n terms-that-are-categories-of-ascents; or, Euler ascents multiply n neighboring figurate amounts which then add to equal a value in a power-of-n series.  The level of figurate numbers involved indicates an attained level of recursive accumulation.  By applying Euler row n to terms in a figurate level, the amounts of new organizations of sets in that set-size at the x^th case level, and the previous (n-1) case levels, thoroughly define a value in a power of n series.  The value may be a shell of shells, a nexus number, an xⁿ value, a sum of the 1^st through x^th of xⁿ values, or sums-of-sums values—depending upon which figurate series of Pascal’s Triangle is weighted by Euler’s row n.  (See figure for a depiction of examples of figurate numbers variously affected by Euler numbers.)  Very interestingly, all series’ values determinable by this Euler/Pascal relationship are contained in the Table 1 pattern even though that table was generated as simply accumulation from Euler “seed numbers”.

Cubic Examples

For example, a cube is about sets of size four or the 1st through xth total sets of size three—the largeness of value of the cube is dependent on how late the case under consideration is, or, in other words, how large of a set from which permutations will be made.  Three neighboring size-4 sets (representing sum total sets of size-3) are weighted by Euler Triangle’s row 3, (which means multiplied—in order of their occurrence,) by the possible orderliness as described by Euler’s row 3, and then added.

A summation-of-cubes is about size-5 sets or the 1st through xth total sets of size four—the largeness of value will depend on how far out is the case.  Size-5 sets (representing sum total sets of size-4) are then parsed by Euler Triangle row 3 into a first term with 1 order, a second neighboring term with 4 orders, and a third neighboring term with 1 order, and then added.

Square Examples

Squares are about size-3 sets or the 1st through xth total sets of size two—largeness of value being dependent on how far out the case—but the size-3 sets (representing sum totals of size-2 sets) are established by Euler row 2 as two neighboring numbers, each having simply one order and then added together.  A summation-of-squares is about size-4 sets—largeness of value being dependent on how far out the case—two neighboring numbers have one order each which are added.  A summation-of-summations-of-squares is about size-5 sets—again, largeness of value being dependent on the extent of the case—and, again, two neighboring numbers have only one order each are added together.

Power of 4 Examples

A number within the power of 4 is about size-5 sets or the 1st through xth total sets of size four—largeness of value depending upon the extent of the case.  Size-5 sets (representing the sum total of size-4 sets) will parse by Euler Triangle row 4 into:  a first term with 1 order, the next term with 11 orders, the third term with 11 orders, and the last term with 1 order.  A summation-of-the-power-of-4 is about size-6 sets—largeness of value depending upon how far out the case—and the size-6 sets parse by Euler’s row 4 into four terms of:  1 order, 11 orders, 11 orders, and 1 order.  A summation-of-summations-of-the-power-of-4 is about size-7 sets—largeness of value depending upon how far out the case.  Size-7 sets parse by Euler row 4 into four terms of:  1 order, 11 orders, 11 orders, and 1 order to define a value that is a summation-of-summations-of-x⁴.

Considering that each power of n is apparently "about" the count of sized-n sets, it is intriguing to look at  the powers' pattern of accumulation and notice that, there, the power of n  begins with sets of n ones which accumulate . . . so this type of accumulation to the powers are also about sets of size n.

Figure 4.  Here are formulas for one layout of the Euler/Pascal cube.  Extending backward would be the value of r accumulations.  Extending rightward would be the value of x.  Each layer of the cube may be considered a power-level of n and is listed below--one equation per power level each of which solves for shells of shell series through power series through summations of summations of powers series depending on r.  Notice that coefficients to the binomial variables are Euler's Triangle.

Figure 5.  Sets of (recognizable) formulas arranged by sheets of a Euler/Pascal cube layout.  Each 'wall' of formulas is at one accumulation level of either shell to powers, powers, or summation of powers and shows parallel formulas across power levels. 

CONCLUSION

There is powerful sense in examining figurate content of recursive accumulations.  In their correspondence, Fermat and Pascal discuss figurate numbers and tend to express thinking of the type described here.  Pascal’s promise in the preface to his 1654 treatise on the Sums of Numerical Powers reads as if hinting about figurate content of powers:

“Given, starting with the unit, some consecutive numbers, for example 1, 2, 3, 4, one knows, by the methods the Ancients made known to us, how to find the sum of their squares, and also the sum of their cubes; but these methods, applicable only to the second and third degrees, do not extend to higher degrees.  In this treatise, I will teach how to calculate not only the sum of squares and of cubes, but also the sum of the fourth powers and those of higher powers up to infinity;  And that, not only for a sequence of consecutive numbers beginning with the unit, but for . . .”

According to David Pengelley who presents cited source material in his paper,  “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).”

The Euler/Pascal organization of the dimensions of powers is a lovely (and powerful) pattern.

Figure 6.  A fundamental relationship of figurate series of Pascal’s Triangle to shell-of-shell series, shell/nexus number series, and power series is diagrammed with Euler Triangle “influence” joining the series together.  The diagrammed relationships continue leftward to earlier difference series and rightward to later summations of summations series.

Click for discussion of Fermat’s Last Theorem in light of the Euler/Pascal relationship.

Cecilia

            September 2003 through August 2004

Send mail to Cecilia@noticingnumbers.net with questions or comments about this web site.
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Last modified: 12/16/05