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User:Peter Luschny/TheLostPascalNumbers

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The Lost Pascal Numbers

The Pascal triangle
               1              
             1    1            
          1    2   1          
         1   3   3    1        
       1   4    6   4    1      
     1    5   10   10    5    1    
   1    6   15   20    15    6    1  
 1   7   21   35   35   21   7    1

Last month I started the blog by looking at the Pascal triangle and asked: Is there a sensible way to fill the holes of the central column 1, *, 2, *, 6, *, 20, *, ... I then went on to suggest the sequence 1, 2, 6, 6, 30, 20, 140, ...

This month I want to go a step further. Of course there are many more `holes´ in the triangle. (Perhaps `Pascal's strainer´ would be a better name than `Pascal's triangle´?) So let us fill them all this time. Here is my suggestion:

The extended Pascal triangle
               1              
             1 1  1            
          1 2  2 2 1          
         1 3 3 6 3 3  1        
       1 4 4 12  6 12 4 4  1      
     1 5  5 20 10 30 10 20  5 5  1    
   1 6  6 30 15 60 20 60  15 30  6 6  1  
 1 7 7 42 21 105 35 140 35 105 21 42 7 7  1

Let us first state the two simple rules of construction:

a b c
  a + c  
a b c
  a + b + c  

The first observation is that our extension is compatible with our extension of the central column suggested last month. We recall that the extended central column is the swinging factorial; thus we see how the swinging factorial is embedded into a larger framework. 

Now  let us turn the tables on Pascal and look what happens if we throw out his contribution to the extended triangle.

The denominators in Leibniz's harmonic triangle
                             
              1              
            2   2            
          3   6   3          
        4   12   12   4        
      5   20   30   20   5    
    6 30 60 60 30 6  
7   42   105   140   105   42   7  

Thus we might say: The complementary Pascal numbers are the denominators in Leibniz's harmonic triangle! (For convenience let us call this triangle in short Leibniz triangle.)

We can also say that the extended binomial triangle is the sum of the Pascal triangle and the Leibniz triangle provided we remember the `holes´ and interpret them as zeros. Thus the long version of our definition is: The sum of the aerated Pascal triangle and the aerated Leibniz triangle is the extended binomial triangle. Why not call it the Pascal—Leibniz triangle?

It also sheds light on the significance of the swinging factorial: having both Pascal and Leibniz as mental parents is a very good omen for a child.

A first summary:

  Triangle Row Sums Central
Pascal A007318 A000079 A000984
Leibniz A003506 A001787 A002457
Pascal—Leibniz A162246 A001792 A056040

Much more is true. The triangle has a beautiful analytic extension as a complex function. Below are two snapshots taken from [1].

The xy-plane reflects the extended Pascal triangle as seen standing in front of the function. The xz-plane nicely shows the swinging structure of this function (the swinging factorial is here the central line). The plots show the logarithm of the function.

With Maple:

Pascal := (n,k) -> binomial(n,k):
seq(print(seq(Pascal(n,k),k=1..n)),n=1..7);

Leibniz := (n,k) -> k*binomial(n,k):
seq(print(seq(Leibniz(n,k),k=1..n)),n=1..7);

PascalLeibniz := proc(n,k) local Omega;
Omega := n -> (n-(n mod 2))/2;
n!/(Omega(n-k)!*Omega(n+k)!) end:
seq(print(seq(PascalLeibniz(n,m),m=-n..n)),n=0..7);

References

  1. Peter Luschny, Divide, swing and conquer the factorial and the lcm{1,2,..,n}, preprint April 2008.
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