[seqfan] A rectangular array.

Ed Jeffery lejeffery2 at gmail.com
Thu Oct 25 07:18:52 CEST 2012


Hello,

I found two arrays (not in OEIS) that seem interesting enough. But what I
present here is empirical, so I thought I should pass along the ideas to
anyone who would like to do more research, work out the details and submit
the sequences (if they are of any value). To that end:

Define an infinite upper triangular array T(n,k) with the natural numbers
on the main diagonal, alternating ones and zeros in the first row, and with
zeros everywhere else, starting like this:

T =
[
1, 0, 1, 0, 1, ...
0, 2, 0, 0, 0, ...
0, 0, 3, 0, 0, ...
0, 0, 0, 4, 0, ...
0, 0, 0, 0, 5, ...
...
].

The transpose of T could be submitted to OEIS as a triangle read by rows,
viz., T = {{1}, {0,2}, {1,0,3}, ...}, because of its association with the
following.

Index the rows of array T by n = 1, 2, ..., and the columns by k = 1, 2,
..., and consider the m X m matrix A_m with entries [A_m]_{i,j} = T(i,j),
i,j = 1, 2, ..., m.

* It seems that the coefficients of the characteristic polynomial of A_m
are given by the m-th row of A008276 [1] (Stirling numbers of the first
kind). *

(In fact, for * there is a comment, stated without proof, in A008276 to
that effect; the proof seems difficult to me, and I would like to know it.)

Now let m become large, and let A = A_m. Then the sequence of first rows
produced by A^N, N = 1, 2, ..., can be put into an array L' with entries
L'(n,k) = [A^n]_{1,k}. The array L' begins as

L' =
[
1, 0,   1, 0,   1, 0,    1, ...
1, 0,   4, 0,   8, 0,   10, ...
1, 0,  13, 0,  31, 0,   57, ...
1, 0,  40, 0, 156, 0,  400, ...
1, 0, 121, 0, 781, 0, 2801, ...
...
].

Form a new array L from L' by eliminating the columns of L' with even index
(because they are all zeros), by defining L(n,k) = L'(n, 2*k-1), while
remembering the association of the new shifted columns with the previous
odd indices (k = 1, 3, 5, ...). The array L then begins as:

L =
[
1,   1,   1,    1,    1, ...
1,   4,   6,    8,   10, ...
1,  13,  31,   57,   91, ...
1,  40, 156,  400,  820, ...
1, 121, 781, 2801, 7381, ...
...
].

The row sequences in L, with the exception of the first three rows, do not
seem to be in OEIS. Row 1 is obvious, and row 2 is evidently A105360 =
{1,2,4,6,8,...} [2] but is missing the 2. Row 3 appears to be A054554 =
{1,3,13,31,57,...} [3] but is missing the 3. So I wonder if columns 2,3,...
could be shifted to the right one place and a new second column =
{1,2,3,...} inserted without ill effect. Would this make the row generating
functions easier to determine explicitly? I ask because of the following:

** It seems that the generating function for row n of L is of the form
F_n(x) = G_n(x)/(1-x)^n. However, I could not find a pattern for the
coefficients in the G_n, since the polynomials seem to be generally
irreducible. **

However:

*** The column sequences in L, for k = 2, 3, ..., seem to be given by
{(3^n-1)/2}, {(5^n-1)/4}, {(7^n-1)/6}, ..., which, from the original
definition of L', leads to an equation for the array given by

L(n,k) = if[k=1, 1, ((2*k-1)^n-1)/(2*(k-1))]. (I could not find a way to
include column 1 otherwise.) ***

**** Because of ***, it seems that column 2 of L is A003462 [4], column 3
is A003463 [5] and column 4 is A023000 [6], etc., and there are likely many
more cross references to be found. ****

Finally, I could not find any of the diagonal sequences in OEIS.

LEJ

References:

[1] A008276, http://oeis.org/A008276
[2] A105360, http://oeis.org/A105360
[3] A054554, http://oeis.org/A054554
[4] A003462, http://oeis.org/A003462
[5] A003462, http://oeis.org/A003463
[6] A023000, http://oeis.org/A023000



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