The Pattern
$$\begin{array}{} +1\\ -1\\ +3\\ \end{array}$$ $$\begin{array}{} +5 & \color{black}{-2} \\ -21 & \color{black}{+4} \\ +99 & \color{black}{-10} \\ \end{array}$$ $$\begin{array}{} +693 & -52 & \color{black}{+6} \\ -3627 & +196 & \color{black}{-14} \\ +18315 & -744 & \color{black}{+36} \\ \end{array}$$ $$\begin{array}{} +290615 & -9370 & +384 & \color{black}{-20} \\ -1614297 & +41884 & -1338 & \color{black}{+50} \\ +8589063 & -183172 & +4686 & \color{black}{-132} \\ \end{array}$$ $$\begin{array}{} +241474401 & -4317040 & +92136 & -2372 & \color{black}{+70} \\ -1387979425 & +21071596 & -376194 & +7930 & \color{black}{-182} \\ +7663714425 & -100027800 & +1515828 & -26640 & \color{black}{+490} \\ \end{array}$$ $$\begin{array}{} ? & ? & ? & ? & ? & ? \\ \end{array}$$ $$ \dots $$
Notice we have "sections" of three rows per section.
Observations so far
Notice the last column from the pattern, if broken into three sequences, is giving the next sequence from the sum of consecutive elements of the previous one:
$$-2, +6, -20, +70, \dots$$ $$+4, -14, +50, \dots$$ $$-10,+36,\dots$$
That is,
Let $f(n)=(-1)^n\binom{2n}{n}$, where $\binom{\square}{\square}$ is the binomial coefficient.
I noticed the last column in $i$-th section, $i=1,2,3,\dots$, is given by: (first section being $0$-th):
$$ f(i)=-2,+6,-20,+70,\dots$$ $$ f(i+1)+f(i)=+4,-14,+50,-182,\dots$$ $$ f(i+2)+2f(i+1)+f(i)=-10,+36,-132,+490,\dots$$
I'm not sure if the other coefficients (terms), from other columns, follow a similar pattern.
My question is, can we determine coefficients in the above pattern for $n$-th row, $n\in\mathbb N$?
That is, can we spot patterns for coefficients in other columns, similar to the one I noticed?
Can we conjecture the pattern from the data, is what I'm asking.
The rest of this post below, is additional info that may or may not help answer my question above.
Context of the pattern
The pattern above is the $\color{red}{red}$ diagonal coefficients from the generalized table (click to zoom in).
Noticed below $A_i$; there is a relation to sums of binomial coefficients, so that might be of use.
For the generalized table patterns, we are looking at the coefficients in digits of $d=2n-1$ digit double palindrome, when expressed as a linear combination of other digits plus some parametric value $c$.
A double palindrome is a number palindromic in number bases $b,b-1$ (two consecutive number bases). Palindromic means its digits are the same when reversed in those bases.
The generalized pattern is a consequence of the following system ($b,a_i,A_i\in\mathbb N_0,i=1,\dots,d$):
$$\sum_{i=1}^{d} a_i b^{d-i}=\sum_{i=1}^{d} A_i (b-1)^{d-i}$$ $$0\le a_i\lt b,0\le A_i\lt b-1,a_1\ne 0,A_1\ne 0$$ $$a_{i}=a_{d-i+1},A_{i}=A_{d-i+1}$$
Where we can show that ($o_i\in\mathbb N_0$):
$$A_i=\sum_{k=1}^{i}\binom{d-k}{i-k}a_k + o_{i} - o_{i-1} (b-1)$$
By expanding $((b-1)+1)^{d-i}$ by binomial theorem. The $o_i$ are chosen such that the inequalities are satisfied, and give a system of linear Diophantine equations for every valid set of $o_i$ parameters, whose order depends on $d$.
The $d$ corresponds to the $(2n-1)$-th row: $d=2n-1$.
I was only been able to partially solve this with a computer, obtaining the pattern so far that way.
Red pattern (main diagonal) given at the beginning
For every solution $(a_1,\dots,a_d;b)$ for given $d$, the $a_{(d+1)/2}$ is the middle digit of a double palindrome and is expressible as a linear combination of $\alpha_i a_{2i-1}$ and some other value $c$ that depends on base, when looked at in base $b-1$.
The coefficient $\alpha_i$ are the numbers in the pattern corresponding to the $d=2n-1$ case/row. For example, if you look at seventh row, $n=7$, it gives $d=13$ and:
$$ a_{7} = +693a_1−52a_3+6a_5\pm c$$
Where $+693,-52,+6$ are given in the seventh row of the pattern at the beginning.
Where $c$ is some function depending on the number base $b$ and some parameters $c_i\in\mathbb N$. If you want to see examples of $c$'s for $d=5$ in terms of $c_1=x,c_2=y,b$, there is a table contained in this answer to a related question.
The generalized pattern
The above pattern in question, is just the main diagonal of the following table, colored in $\color{red}{red}$:
For Column patterns, I found: (Thanks to user TheSimpliFire for $C^5$)
$$ \color{darkred}{C^1_d(a_1)=-\frac{1}{2}\binom{d-1}{ 1} =\dots,-2,-3,-4,-5,\dots} \\ \color{green}{C^2_d(a_1)=\frac{1}{4}\binom{d-1}{ 3} =\dots,30,55,91,140,\dots} \\ \color{darkblue}{C^3_d(a_1)=-\frac{1}{2}\binom{d-1}{ 5} =\dots,-2184,-4284,-7752,-13167,\dots} \\ \color{purple}{C^4_d(a_1)=\frac{17}{8}\binom{d-1}{ 7} =\dots,362406,735471,1397825,2516085\dots} \\ \color{brown}{C^5_d(a_1)=-\frac{31}{2}\binom{d-1}{ 9}} =\dots,\color{orange}{−48430525,}\color{brown}{- 107056950,\dots} \\ $$
Also notice that $C$ patterns from $a_1$ follow in all other $a_3,a_5,\dots$ but shifted. I don't believe $C^6$ can be seen unless I manage to crack some more terms (rows) for the table.
I'm not sure about any of the diagonal patterns, similar to the main one presented at the beginning. Except, for the coefficients in the last column: - For other diagonals as well, last column coefficients follow a similar pattern to the one presented at the beginning. But I do not see patterns for any other coefficients on diagonals, hence this question.
The pattern I'm asking about in this question is the red diagonal as presented at the beginning, hoping it will shed some light on other diagonals as well. - I want to completely determine this entire table for any $d$ eventually.
Follow up question: - Can we determine coefficients on any other diagonals as well, for every $d$?
(The pattern at the beginning represents the red, main diagonal, and is the main focus of this question. - If a pattern there can be seen, similar one should follow in other diagonals.)
Motivation
This will help solve the following questions:
- Arbitrarily long palindromes in two consecutive number bases
- Find palindromes in two consecutive number bases?
- Can long numbers be “3-palindromic”?
- Can a number be palindrome in 4 consecutive number bases?
- Hard system in integers related to natural number representations
And a couple more similar but a bit less related questions.
These patterns for linear combinations from the generalized table, along with patterns for $c_0,c$ expressions, will completely determine all double palindromes .