Background:

A050447 http://oeis.org/A050447

Table T(n,m) giving total degree of n-th-order elementary symmetric polynomials in m variables, -1 <= n, 1 <= m, transposed and read by upward antidiagonals.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 5, 1, 1, 5, 10, 14, 8, 1, 1, 6,15

A007318 http://oeis.org/A007318

Pascal’s triangle read by rows: C(n,k) = binomial(n,k) = n!/(k!*(n-k)!), 0 <= k <= n.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 6, 15,

Question I:

You can see by inspection of:

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 5, 1, 1, 5, 10, 14, 8, 1, 1, 6,15

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 6,15

that (1,6,15) occurs in corresponding positions in the two sequences.

Is this a coincidence?

I ask for two reasons.

First, John H. Conway once said to me, “The Devil bears gifts also”, and ever since then, I have sought to rule out possible coincidences FIRST, BEFORE wasting time in pointless investigations.

Second, if this is NOT a coincidence, then:

1) I really need an explanation of WHY it’s not, because (1,6,15) show up quite dramatically in a possible bio-molecular realization of the root-system of $ E_8$ .

2) I need to know how to clear the 4th, 5th, and 6th terms (with respective co-efficients 20, 15, 6) in the binomial expansion for n = 6, because if there is some set of rotations that will do this, then the result will be terms with co-efficients (1,6,15,1), and this is the set I really need, not merely (1,6,15).

Thanks as always for whatever time you can afford to spend considering this matter.