R J Cano on Thu, 31 Jan 2013 00:29:41 +0100

 19, 36, 51, 64, 75, 84, 91, 96, 99, 100, 99, 96, 91, 84, 75, 64, 51, 36, 19

--------------------------------------------------------------------
Actual subject: On the true nature of A215940
--------------------------------------------------------------------

Greetings,

An interesting observation:

19, 36, 51, 64, 75, 84, 91, 96, 99, 100, 99, 96, 91, 84, 75, 64, 51, 36, 19

This is a symmetric sequence. The OEIS server replies:

---------------------------------------------------------------------------------

Sorry, but the terms do not match anything in the table.
Your sequence appears to be: −1*x^2 + 20*x

---------------------------------------------------------------------------------

Upon the interesting Diophantine F(k)=−1*k^2 + 20*k
recreating such sequence, these numbers are the
coefficients for the polynomial g(r) in r=20 that gives the
last one inside the first 20! terms of A215940.

g(r)= sum_{u=1..r} F(u-1)*r^(r-u)

g(20)= 5488245866744423385333139.

What is interesting here is not big quotient described by g(20),
but the following (observed) fact:

There exists <--- for the last element inside each set with the first N!
terms of A215940 ---> a symmetric set of coefficients for the polynomial
that describes it in a base independent way.

This behavior suggest additionally to treat the quotients defined in A215940
as vectors in the sense of the Tensor Algebra and Calculus used in Physics
( I mean base independent for the present context if it where allowed the
analogy between the radices for the positional systems and the coordinate
systems ).

Therefore it must be definable a sort of vector space equipped with
the permutations without repetitions as its elements and a proper definition
of inner product which yields the quotients A215940 explaining why they
are invariant.

A215940 actually is a tensor problem.

Cheers.

R. J. Cano

P.S.: These are the first 20 sets where it have sense to define them.

0
1
2,2
3,4,3
4,6,6,4
5,8,9,8,5
6,10,12,12,10,6
7,12,15,16,15,12,7
8,14,18,20,20,18,14,8
9,16,21,24,25,24,21,16,9
10,18,24,28,30,30,28,24,18,10
11,20,27,32,35,36,35,32,27,20,11
12,22,30,36,40,42,42,40,36,30,22,12
13,24,33,40,45,48,49,48,45,40,33,24,13
14,26,36,44,50,54,56,56,54,50,44,36,26,14
15,28,39,48,55,60,63,64,63,60,55,48,39,28,15
16,30,42,52,60,66,70,72,72,70,66,60,52,42,30,16
17,32,45,56,65,72,77,80,81,80,77,72,65,56,45,32,17
18,34,48,60,70,78,84,88,90,90,88,84,78,70,60,48,34,18
19,36,51,64,75,84,91,96,99,100,99,96,91,84,75,64,51,36,19

This is the true kind of "Universally invariant" look of the terms (*) for A215940!!

First consequence. Property: The greatest coefficient in each one
of those sets is the smallest radix from which all the terms looks
unchanged under radix conversion, referring here to the first (m+1)!
terms of A215940 where m is the first coefficient from left to right
in the corresponding row.

---------
(*) f.n.: Placed at exact factorial offsets.