Talk:Regular orthotopic numbers

Let me comment on the section "Procedure to find the generating functions for successive powers".

What is done here is a slight variation of what Leonhard Euler did in 1755. Look at the facsimile on Eulerian numbers.

So your generating function is G(n,x) = x*E(n,x)/(1-x)^(n+1) where E(n,x) are the Eulerian polynomials.

Thus the 'procedure' boils down to a recursion of the triangle of Euler, which is described on A008292 as

 EulerianNumber := proc(n,k) 
                     if k < 1 or k > n then 
                       0 
                     elif n = 1 then 
                       1 
                     else 
                       (n-k+1)*EulerianNumber(n-1,k-1)+k*EulerianNumber(n-1,k) 
                     fi 
                   end;

 EulerianPoly := proc(n,x) 
                   add(EulerianNumber(n,k)*x^(k-1),k=1..n) 
                 end;

So we have a full description of the generating function:

 GF_RegularOrhtotopicNumbers := proc(n,x) 
                                  x*EulerianPoly(n,x)/(1-x)^(n+1) 
                                end;

Perhaps this might save one or the other of the formulas. I am personally always scared when I see so many formulas because it makes me think that the matter is very complicated and I will never be able to understand it.

Peter Luschny 20:57, 21 July 2010 (UTC)

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Thanks for this cool information!

Thanks for this cool information! I created the page Eulerian numbers, triangle of, check out the row sums!

Daniel Forgues 22 July 2010

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Hi Daniel, with the definition of the Eulerian polynomials according to Knuth...

Hi Daniel, with the definition of the Eulerian polynomials according to Knuth (and to http://dlmf.nist.gov/) things became simpler! Look at the Eulerian polynomials page and observe that the 'generating function of regular orthotopic numbers' now *is precisley* what Euler wrote in 1749 (we can now drop the factor 'x').

I will show it with Maple: Eulerian Polynomials as defined by Knuth:

 E := proc(n,x) 
        if n = 0 then 
          1 
        else
          x*(1-x)*diff(E(n-1,x),x)+E(n-1,x)*(1+(n-1)*x)
        fi 
      end:

 seq(print(sort(expand(E(n,x)))),n=0..5);

              1
              1
            x + 1
         x^2  + 4 x + 1
    x^3  + 11 x^2  + 11 x + 1
 x^4  + 26 x^3  + 66 x^2  + 26 x + 1

Eulerian Polynomials as defined by Euler:

 G := proc(n,x) 
        E(n,x)/(1-x)^(n+1)  
      end:

And this gives us now all your tables:

 seq(print([n^k],sort(factor(G(k,x))),coeffs(convert(series(G(k,x),x),polynom))),k=0..5);

          1
 [1], - -----, 1, 1, 1, 1, 1, 1
        x - 1

         1
 [n], --------, 1, 2, 3, 4, 5, 6
             2
      (x - 1)

   2      x + 1
 [n ], - --------, 1, 4, 9, 16, 25, 36
                3
         (x - 1)

        2
   3   x  + 4 x + 1
 [n ], ------------, 1, 8, 27, 64, 125, 216
               4
        (x - 1)

                   2
   4     (x + 1) (x  + 10 x + 1)
 [n ], - -----------------------, 1, 16, 81, 256, 625, 1296
                      5
               (x - 1)

        4       3       2
   5   x  + 26 x  + 66 x  + 26 x + 1
 [n ], -----------------------------, 1, 32, 243, 1024, 3125, 7776
                        6
                 (x - 1)

Peter Luschny 00:37, 24 July 2010 (UTC)

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I observed throughout OEIS and MathWorld that the convention for figurate numbers is that it is 0 for n = 0 and 1 for n = 1...

I observed throughout OEIS and MathWorld that the convention for figurate numbers is that it is 0 for n = 0 and 1 (initial dot) for n = 1 while the convention for GLOBALLY (e.g. "centered" pyramidal numbers are NOT globally centered, it is only a stacking of centered polygons and starts at 0 for n = 0 since it does not have a global central dot) centered figurate numbers is to have the central dot, giving 1, for n = 0. To respect that convention I need to multiply the generating functions by x for the regular orthotopic numbers:

 
        x
 [1], -----, 0, 1, 1, 1, 1, 1, 1
      1 - x

         x
 [n], --------, 0, 1, 2, 3, 4, 5, 6
             2
      (1 - x)

   2   x (x + 1)
 [n ], ---------, 0, 1, 4, 9, 16, 25, 36
              3
       (1 - x)

           2
   3   x (x  + 4 x + 1)
 [n ], ----------------, 0, 1, 8, 27, 64, 125, 216
                  4
           (1 - x)

                   2
   4   x (x + 1) (x  + 10 x + 1)
 [n ], -------------------------, 0, 1, 16, 81, 256, 625, 1296
                    5
             (1 - x)

           4       3       2
   5   x (x  + 26 x  + 66 x  + 26 x + 1)
 [n ], ---------------------------------, 0, 1, 32, 243, 1024, 3125, 7776
                        6
                 (1 - x)

Daniel Forgues 24 July 2010

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I see. This is a good point. However note that for example in Wikipedia's table these sequences all start with n=1.

Daniel: "I observed throughout OEIS and MathWorld that the convention for figurate numbers is that it is 0 for n = 0..."

I see. This is a good point. However note that for example in Wikipedia's table these sequences all start with n=1. http://en.wikipedia.org/wiki/Polygonal_number#Table_of_values It seems to me that the case 'n=0' is not looked upon as counting a regular geometric pattern by most authors.

Peter Luschny 08:57, 24 July 2010 (UTC)

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In the OEIS, the figurate numbers sequences authored by Neil Sloane typically start at n = 0 giving 0 followed by n = 1 giving 1, ...

In the OEIS, the figurate numbers sequences authored by Neil Sloane typically start at n = 0 giving 0 followed by n = 1 giving 1, while the GLOBALLY centered figurate numbers start at n = 0 giving 1 for the mandatory central dot. Sequences authored by other people do not always follow that convention. That convention seems the most common. It seems to me that to always have 1 (the initial dot for figurate numbers; the globally central dot for GLOBALLY centered figurate numbers) for n = 0 would have been a better convention since n would then correspond to the number of nondegenerate subfigures.

Just check the OEIS sequences of figurate numbers and GLOBALLY centered figurate numbers, especially the ones authored by Neil Sloane.

For figurate numbers, even if we skip n = 0 but still have the initial dot for n = 1, the generating function must associate a(1) = 1 with x^1 in its series representation, so we still need need to multiply with x, otherwise the 1 will be the constant term, i.e. associated with x^0 thus a(0). Only if we had a(0) = 1 could we spare multiplying the generating function by x.

For example, for the cubic numbers, we want the mapping:

1 --> 1, 2 --> 8, 3 --> 27, ..., n --> n^3

so the generating function must start: (Cf. $\scriptstyle {\frac {x(1+4x+x^{2})}{(1-x)^{4}}}\,$, Wolfram Alpha.)

x + 8 x^2 + 27 x^3 + ...

If we had the mapping:

0 --> 1, 1 --> 8, 2 --> 27, ..., n --> (n+1)^3

then the generating function would start: (Cf. $\scriptstyle {\frac {(1+4x+x^{2})}{(1-x)^{4}}}\,$, Wolfram Alpha.)

1 + 8 x + 27 x^2 + ...

In the following, EulerianNumber(n,n) with n ≥ 1 will be set to 0, as it should be, from the recursion rule!

 EulerianNumber := proc(n,k) 
                     if k < 0 or k > n then 
                       0 
                     elif n = 0 then 
                       1 
                     else 
                       (n-k)*EulerianNumber(n-1,k-1)+(k+1)*EulerianNumber(n-1,k) 
                     fi 
                   end;

 EulerianPoly := proc(n,x) 
                   add(EulerianNumber(n,k)*x^k,k=0..n) 
                 end;

Generating function:

 GF_RegularOrhtotopicNumbers := proc(n,x) 
                                  x*EulerianPoly(n,x)/(1-x)^(n+1) 
                                end;

Daniel Forgues 24 July 2010

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I realize what an impressive work you add to the wiki. However, ...

However, I think such a work has to be based on a clear definition of the subject. An argumentum ad verecundiam cannot be a substitute for that.

Moreover, the core sequences on OEIS have to cover more than one interpretation. If I were to write a sequence like A000290 I would not make it differently. This does not imply that I am forced to think that the square numbers in the sense of 'figurative numbers' have to start at n = 0. For example this would be the case if I use a definition similar to: "A figurative number is a number which counts figures. A figure is a nonempty set of points arranged in some geometric pattern ... etc."

By the way, I looked up the "Book of Numbers" by John H. Conway and Richard K. Guy., Chapter 2, "Figures from figures. Doing arithmetic and algebra by geometry". Looking at the many examples given, I see that *all* start at n = 1.

And I am well aware of the wisdom of N.J.A. Sloane who once said:

"Start a sequence at the earliest reasonable point, but no earlier."

Peter Luschny 09:16, 25 July 2010 (UTC)

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"Start a sequence at the earliest reasonable point, but no earlier." is a cool adaptation of Einstein's saying...

"Start a sequence at the earliest reasonable point, but no earlier." is a cool adaptation of Einstein's saying... Does it imply that the starting point of a sequence is relative to the frame of reference? ;-)

Wether we consider the empty figurate number (with 0 dots for n ≤ 0) as a figurate number or not is a matter of convention, like wether we consider the empty set as a set or not... or 0 as a natural number or not (which is sometimes considered natural, sometimes unnatural?... it was not even a number for the Ancient Greeks...)?

A bit like it is convenient to define 0^0 = 1 and 0! = 1 for consistency of the binomial expansion formula for polynomials, for many equations relating figurate numbers, it is often convenient to have figurate numbers to be defined as 0 for n ≤ 0, e.g. for n = 1 or 2 in the following:

$P_{6}^{(3)}(n)=P_{4}^{(3)}(n)+2\ P_{4}^{(3)}(n-1)+P_{4}^{(3)}(n-2),\,$

which gives octahedral numbers (6 vertices) in terms of tetrahedral numbers; or for n = 1 in the following:

$P_{4}^{(2)}(n)=P_{3}^{(2)}(n)+P_{3}^{(2)}(n-1),\,$

which gives square numbers in terms of triangular numbers.

There are very many figurate numbers equations of this type which beg for the convention that for n ≤ 0 we should have 0, the empty figurate number... (so to speak)

Anyway, whether or not we consider 0 for n = 0 does not impact the generating function since we still keep 1 for n = 1 (so there was no shifting, there would be a shifting if we had 1 for n = 0 were the generating would be divided by x.)

Since most figurate numbers sequences in the OEIS start with 0 for n = 0 (while the globally centered figurate numbers starting with 1 for n = 0) I made the sequences in the wiki to abide by the same. I feel like leaving it as it is for now and ask for advice from Neil Sloane and other people before removing all entries for n = 0 for the not globally centered figurate numbers (I don't want to be told later that I should not have removed those and to have to put them all back again...) Removing them would make the wiki less consistent with the OEIS Classic... I'll ask Neil Sloane about it...

Thanks

Daniel Forgues 25 July 2010

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If you look at (k,1)-Pascal (rectangular) triangle columns and (k+2)-gonal (hyper)pyramidal numbers, ...

If you look at (k,1)-Pascal (rectangular) triangle columns and (k+2)-gonal (hyper)pyramidal numbers, it seems that the convention giving the most symmetry in the triangle would be to have 1 (for the initial dot) for n = 0. With this convention, n would indicate the number of nondegenerate subfigures of the figurate number, e.g.:

• For triangular numbers, we would get:

$P_{3}^{(2)}(0)=1,\ P_{3}^{(2)}(1)=3,\ P_{3}^{(2)}(2)=6,\ P_{3}^{(2)}(3)=10,\ \ldots \,$

• For square numbers, we would get:

$P_{4}^{(2)}(0)=1,\ P_{4}^{(2)}(1)=4,\ P_{4}^{(2)}(2)=9,\ P_{4}^{(2)}(3)=16,\ \ldots \,$

For example, with this convention $P_{3}^{(2)}(3)=10\,$ is the third nondegenerate triangular number:

   x
  / \
  x-x
 /   \
 x-x-x
/     \
x-x-x-x

and $P_{4}^{(2)}(3)=16\,$ is the third nondegenerate square number:

x-x-x-x
| | | |
x-x x x
|   | |
x-x-x x
|     |
x-x-x-x

Unfortunately?, this is not the followed convention...

Daniel Forgues 17:20, 29 July 2010 (UTC)

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(It's in the bible: Am Anfang war die Welt wuest und leer, therefore a(0)=0 :-).

From User:Richard Mathar on seqfan:

(It's in the bible: Am Anfang war die Welt wuest und leer, therefore a(0)=0 :-).

From: Richard Mathar <mathar@strw.leidenuniv.nl> To: seqfan@seqfan.eu Date: Thu, 29 Jul 2010 16:10:05 +0200 Subject: [seqfan] Re: "Start a sequence at the earliest reasonable point, but no earlier."

http://list.seqfan.eu/pipermail/seqfan/2010-July/005448.html spake

df> In the OEIS, the figurate numbers sequences authored by Neil Sloane df> typically start at n = 0 giving 0 followed by n = 1 giving 1, while df> the GLOBALLY centered figurate numbers start at n = 0 giving 1 for the df> mandatory central dot.

Many of these sequences have simple polynomial representations. If one of the factors is a 1-st order polynomial [example (3+n)*(n^3-4*n^2+10*n-3) ] it is generally preferable to re-number this by the shift 3+n -> n which changes the offset and maps the 1-st order polynomial to n. Generally this normalizes the polynomial sequences to a(0)=0 by convention, not referring to counts of some sort. (It's in the bible: Am Anfang war die Welt wuest und leer, therefore a(0)=0 :-).

RJM

Message copied from seqfan by:

Daniel Forgues 21:19, 30 July 2010 (UTC)

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What I am still missing is a definition. Before a definition is not set I do not know...

df>  Wether we consider the empty figurate number (with 0 dots for n = 0) 
 as a figurate number or not is a matter of convention, 

What I am still missing is a definition. Before a definition is not set I do not know what we are talking about and cannot decide what convention on top of that definition is useful.

 df>  .. like wether we consider the empty set as a set or not... 

Certainly not. The empty set is a set. Otherwise you would break ZF set theory and all of modern mathematics.

 df>  .. or 0 as a natural number or not (which is sometimes considered natural, 
 sometimes unnatural?...  

Therefore the concept of 'natural number' is not a mathematical one. In mathematics a certain model of Peano arithmetic is given the name N. Look here on Wikipedia:

 It should be noted that the "0" in the above definition need not 
 correspond to what we normally consider to be the number zero. 
 "0" simply means some object that when combined with an appropriate 
 successor function, satisfies the Peano axioms. All systems that 
 satisfy these axioms are isomorphic, the name "0" is used here 
 for the first element, which is the only element that is not a 
 successor. For example, the natural numbers starting with one 
 also satisfy the axioms, if the symbol 0 is interpreted as the 
 natural number 1, the symbol S(0) as the number 2, etc. In fact, 
 in Peano's original formulation, the first natural number was 1.

This illustrates nicely what I mean: that a definition is something different from a convention. First you have to give a definition, then we can settle for convenience on a common way to talk about this definition, i.e. choose a convention.

Note that I tried to give some examples of what might be an appropriate definition in the case under consideration on my user page.

Peter Luschny 09:56, 31 July 2010 (UTC)