Hi Mike -
welcome in the club of pattern-matcher... :-)
I've seen these coefficients in the inverse Schröder-function for the dxp(x)-iterates. I don't have it at hand and hope I recall it correct, but we had a thread in which -I think- Sheldon discussed that somehow complementary coefficients (which occur also in the non-inverse Schröder-function). which were
\( \begin{matrix} {rrrrrrrrrrrrrrr}
. & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\
1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\
-1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\
1 & 2 & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\
-1 & -6 & -5 & -6 & . & . & . & . & . & . & . & . & . & . & . & . \\
1 & 14 & 24 & 45 & 46 & 26 & 24 & . & . & . & . & . & . & . & . & . \\
-1 & -30 & -89 & -214 & -374 & -416 & -511 & -461 & -330 & -154 & -120 & . & . & . & . & . \\
1 & 62 & 300 & 889 & 2177 & 3368 & 5188 & 6980 & 8007 & 7867 & 7188 & 6111 & 4270 & 2528 & 1044 & 720
\end{matrix} \)
to compare to that of yours again:
\( \begin{matrix} {rrrrrrrrrrrrrrr}
. & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\
1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\
1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\
2 & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\
6 & 6 & 5 & 1 & . & . & . & . & . & . & . & . & . & . & . & . \\
24 & 36 & 46 & 40 & 24 & 9 & 1 & . & . & . & . & . & . & . & . & . \\
120 & 240 & 390 & 480 & 514 & 416 & 301 & 160 & 64 & 14 & 1 & . & . & . & . & . \\
720 & 1800 & 3480 & 5250 & 7028 & 8056 & 8252 & 7426 & 5979 & 4208 & 2542 & 1295 & 504 & 139 & 20 & 1
\end{matrix}
\)
I've had the same coefficients in my discussion of the Bell-matrix for dxp(x) (in my then notation Ut(x)) and its symbolic powers.
Here we find that coefficients in the second column (the coefficents of Sheldon) and in the last column (your coefficients) of the A-coefficients-matrix as shown in http://go.helms-net.de/math/tetdocs/APT.htm .
That they are from that columns corresponds to the fact, that the Schröder-function is a scaling of the iterate of (positive) infinite height, so that only that of the second column remain significant, and that of the inverse Schröder-function are that of the negative infinite height (this is just a guess, but accidentally seems to fit) which keep only that of the last column significant, because heights shift the columns down/upwards by construction.
For a sketch see the following. I use the coefficients-matrix A_4 at x^4 according to my mentioned paper. A_4 is
\( \begin{matrix} {rrrr}
. & -1 & 7 & -12 & 6 \\
. & -6 & 18 & -18 & 6 \\
. & -5 & 18 & -18 & 5 \\
. & -6 & 11 & -6 & 1
\end{matrix} \)
and contains your coefficients of n=4 in the last column and the "complementary" in the second column. This is for height h=1. If we have height h=2 we get
\( \begin{matrix} {rrrr}
. & . & . & . & . \\
. & -1 & . & . & . \\
. & -6 & 7 & . & . \\
. & -5 & 18 & -12 & . \\
. & -6 & 18 & -18 & 6 \\
. & . & 11 & -18 & 6 \\
. & . & . & -6 & 5 \\
. & . & . & . & 1
\end{matrix} \)
With increasing height that columns get accordingly shifted. From height h=5 on the columns are strictly"separated", the rowsums consist only on the entries of one column only.
\( \begin{matrix} {rrrr}
. & . & . & . & . \\
. & . & . & . & . \\
. & . & . & . & . \\
. & . & . & . & . \\
. & -1 & . & . & . \\
. & -6 & . & . & . \\
. & -5 & . & . & . \\
. & -6 & . & . & . \\
. & . & 7 & . & . \\
. & . & 18 & . & . \\
. & . & 18 & . & . \\
. & . & 11 & . & . \\
. & . & . & -12 & . \\
. & . & . & -18 & . \\
. & . & . & -18 & . \\
. & . & . & -6 & . \\
. & . & . & . & 6 \\
. & . & . & . & 6 \\
. & . & . & . & 5 \\
. & . & . & . & 1
\end{matrix} \)
In the limit of height infinity only the last column is significant (if the log of the base is u>1) or the second (if the log of the base is u<1), so in the Schröder-function we rediscover only the coefficients of that column. Sorry I've no time to go into more detail here at the moment, but I hope it is understandable so far.
I didn't find any simpler construction for that coefficents than just this - a construction of R. Mathar, which I described elsewhere in the forum used only the Stirlingnumbers but was also iterative and thus had the same complexity for the computation. The coefficients can also be found by solving linear equation-systems using that column-shifting depending on the height-parameter and the hypothesis of even divisibility by the denominator at integer hights. (I did some sketches for that solution but got tired so didn't make it a full explicite procedure)
The relation to the q-analogues is very striking; however precisely this relation makes me skeptical, whether we are on the right track with the powerseries so far since exactly that q-analogues occured in an exercise of L.Euler, where he discussed another series for the log-function and used that q-analogues in the denominator. However: that "log" was correct only on the integer exponents, but was a false log at fractional exponents... a striking analogy to our tetrational problems with fractional heights while integer heights are easy to handle. I discussed this in short with Henryk here in the forum (but without conclusive results); search for the terms "false logarithm" and I think also in sci.math or sci.research.
(I'm currently much occupied with my two statistical courses so please excuse the short style of this msg and the missing references into the msgs of our forum here. Perhaps I can come back to this later in the week. Please ask any specific question anyway)
Gottfried
welcome in the club of pattern-matcher... :-)
I've seen these coefficients in the inverse Schröder-function for the dxp(x)-iterates. I don't have it at hand and hope I recall it correct, but we had a thread in which -I think- Sheldon discussed that somehow complementary coefficients (which occur also in the non-inverse Schröder-function). which were
\( \begin{matrix} {rrrrrrrrrrrrrrr}
. & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\
1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\
-1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\
1 & 2 & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\
-1 & -6 & -5 & -6 & . & . & . & . & . & . & . & . & . & . & . & . \\
1 & 14 & 24 & 45 & 46 & 26 & 24 & . & . & . & . & . & . & . & . & . \\
-1 & -30 & -89 & -214 & -374 & -416 & -511 & -461 & -330 & -154 & -120 & . & . & . & . & . \\
1 & 62 & 300 & 889 & 2177 & 3368 & 5188 & 6980 & 8007 & 7867 & 7188 & 6111 & 4270 & 2528 & 1044 & 720
\end{matrix} \)
to compare to that of yours again:
\( \begin{matrix} {rrrrrrrrrrrrrrr}
. & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\
1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\
1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\
2 & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\
6 & 6 & 5 & 1 & . & . & . & . & . & . & . & . & . & . & . & . \\
24 & 36 & 46 & 40 & 24 & 9 & 1 & . & . & . & . & . & . & . & . & . \\
120 & 240 & 390 & 480 & 514 & 416 & 301 & 160 & 64 & 14 & 1 & . & . & . & . & . \\
720 & 1800 & 3480 & 5250 & 7028 & 8056 & 8252 & 7426 & 5979 & 4208 & 2542 & 1295 & 504 & 139 & 20 & 1
\end{matrix}
\)
I've had the same coefficients in my discussion of the Bell-matrix for dxp(x) (in my then notation Ut(x)) and its symbolic powers.
Here we find that coefficients in the second column (the coefficents of Sheldon) and in the last column (your coefficients) of the A-coefficients-matrix as shown in http://go.helms-net.de/math/tetdocs/APT.htm .
That they are from that columns corresponds to the fact, that the Schröder-function is a scaling of the iterate of (positive) infinite height, so that only that of the second column remain significant, and that of the inverse Schröder-function are that of the negative infinite height (this is just a guess, but accidentally seems to fit) which keep only that of the last column significant, because heights shift the columns down/upwards by construction.
For a sketch see the following. I use the coefficients-matrix A_4 at x^4 according to my mentioned paper. A_4 is
\( \begin{matrix} {rrrr}
. & -1 & 7 & -12 & 6 \\
. & -6 & 18 & -18 & 6 \\
. & -5 & 18 & -18 & 5 \\
. & -6 & 11 & -6 & 1
\end{matrix} \)
and contains your coefficients of n=4 in the last column and the "complementary" in the second column. This is for height h=1. If we have height h=2 we get
\( \begin{matrix} {rrrr}
. & . & . & . & . \\
. & -1 & . & . & . \\
. & -6 & 7 & . & . \\
. & -5 & 18 & -12 & . \\
. & -6 & 18 & -18 & 6 \\
. & . & 11 & -18 & 6 \\
. & . & . & -6 & 5 \\
. & . & . & . & 1
\end{matrix} \)
With increasing height that columns get accordingly shifted. From height h=5 on the columns are strictly"separated", the rowsums consist only on the entries of one column only.
\( \begin{matrix} {rrrr}
. & . & . & . & . \\
. & . & . & . & . \\
. & . & . & . & . \\
. & . & . & . & . \\
. & -1 & . & . & . \\
. & -6 & . & . & . \\
. & -5 & . & . & . \\
. & -6 & . & . & . \\
. & . & 7 & . & . \\
. & . & 18 & . & . \\
. & . & 18 & . & . \\
. & . & 11 & . & . \\
. & . & . & -12 & . \\
. & . & . & -18 & . \\
. & . & . & -18 & . \\
. & . & . & -6 & . \\
. & . & . & . & 6 \\
. & . & . & . & 6 \\
. & . & . & . & 5 \\
. & . & . & . & 1
\end{matrix} \)
In the limit of height infinity only the last column is significant (if the log of the base is u>1) or the second (if the log of the base is u<1), so in the Schröder-function we rediscover only the coefficients of that column. Sorry I've no time to go into more detail here at the moment, but I hope it is understandable so far.
I didn't find any simpler construction for that coefficents than just this - a construction of R. Mathar, which I described elsewhere in the forum used only the Stirlingnumbers but was also iterative and thus had the same complexity for the computation. The coefficients can also be found by solving linear equation-systems using that column-shifting depending on the height-parameter and the hypothesis of even divisibility by the denominator at integer hights. (I did some sketches for that solution but got tired so didn't make it a full explicite procedure)
The relation to the q-analogues is very striking; however precisely this relation makes me skeptical, whether we are on the right track with the powerseries so far since exactly that q-analogues occured in an exercise of L.Euler, where he discussed another series for the log-function and used that q-analogues in the denominator. However: that "log" was correct only on the integer exponents, but was a false log at fractional exponents... a striking analogy to our tetrational problems with fractional heights while integer heights are easy to handle. I discussed this in short with Henryk here in the forum (but without conclusive results); search for the terms "false logarithm" and I think also in sci.math or sci.research.
(I'm currently much occupied with my two statistical courses so please excuse the short style of this msg and the missing references into the msgs of our forum here. Perhaps I can come back to this later in the week. Please ask any specific question anyway)
Gottfried
Gottfried Helms, Kassel
