Taylor polynomial. System of equations for the coefficients.

[marraco]

\( {\color{Red} {

a^{^xa}=\\
\,\,\,\,\,\,\,\,\,\,\,\,a \\
\,+\, lna.a.a_1 \,.\, x \\
\,+\, (lna.a.a_2 + \frac{lna^2}{2}.a.a_1^2) \,.\, x^2 \\
\,+\, (lna.a.a_3 + lna^2.a.a_1.a_2 +\frac{lna^3.a}{6}.a_1^3) \,.\, x^3 \\
\,+\, (lna.a.a_4 + lna^2.a.a_1.a_3 + \frac{lna^2.a}{2}.a_2^2 +\frac{lna^3.a}{2}.a_2.a_1^2+\frac{lna^4.a}{24}.a_1^4) \,.\,x^4 \\
\,+\, (lna.a.a_5 + lna^2.a.a_1.a_4 + {lna^2.a}.a_2.a_3 + \frac{lna^3.a}{2}.a_3.a_1^2 +\frac{ lna^4.a}{6}.a_2.a_1^3+ \frac{lna^3.a}{2}.a_1.a_2^2+\frac{lna^5.a}{120}.a_1^5) \,.\,x^5
\\
\,+\, (lna.a.a_6 + {lna^2.a}.a_1.a_5 + lna^2.a.a_2.a_4 + \frac{lna^3.a}{2}.a_4.a_1^2 +\frac{lna^4.a}{6}.a_3.a_1^3+ \frac{lna^2.a}{2}.a_3^2 + { lna^3.a}.a_1.a_3.a_2 + \frac{lna^3.a}{6}.a_2^3 +\frac{lna^4.a}{4}.a_2^2 . a_1^2+\frac{lna^5.a}{24}.a_2.a_1^4+\frac{lna^6.a}{ 720}.a_1^6) \,.\,x^6 \\
} \)



\( \mathbf{
-The \,\ number \,\ of \,\ terms \,\ for \,\ each \,\ element \,\ in \,\ the \,\ sequence
} \)

This is the number of terms from the first to the -power of x⁹- element: 1, 1, 2, 3, 5, 7, 11, 15, 22, 30

I threw the sequence to The On-Line Encyclopedia of Integer Sequences
http://oeis.org/search?q=1%2C2%2C3%2C5%2...&go=Search

and I got many different ways to calculate it. The first one is: "number of partitions of n (the partition numbers)."


which PariGP can calculate with the numbpart() function:
gp > a=vector(20,k,numbpart(k))
%3 = [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627]

it works up to 56, but I truncated the series, so it should not work beyond that.



\( \mathbf{
-The \,\ index \,\ of \,\ the \,\ coefficients \,\ seems \,\ to \,\ be \,\ also \,\ the \,\ numbers \,\ on \,\ which \,\ n \,\ can \,\ be \,\ partitioned.
} \)

For example, the 4th element, for x⁴, has 5 elements, because 4 can be partitioned in 5 different ways.
For each partition, there is an index i for each number in that partition:

\(
(lna.a.a_4 + lna^2.a.a_3.a_1 + \frac{lna^2.a}{2}.a_2^2 +\frac{lna^3.a}{2}.a_2.a_1^2+\frac{lna^4.a}{24}.a_1^4) \,.\,x^4
\\
\\
4 \right a_4\\
3 + 1 \right a_3.a_1\\
2 + 2 \right a_2^2\\
2 + 1 + 1 \right a_2.a_1^2\\
1 + 1 + 1 + 1 \right a_1^4\\
\)



\( \mathbf{
-The \,\ exponent \,\ of \,\ the \,\ logarithm \,\ lna \,\ is \,\ the \,\ quantity \,\ of \,\ elements \,\ on \,\ each \,\ partition:
} \)

\(
(lna.a.a_4 + lna^2.a.a_3.a_1 + \frac{lna^2.a}{2}.a_2^2 +\frac{lna^3.a}{2}.a_2.a_1^2+\frac{lna^4.a}{24}.a_1^4) \,.\,x^4
\)

\(
4 \right 1 \, elements \right lna \\
3 + 1 \right 2 \,elements \right lna^2 \\
2 + 2 \right 2 \, elements \right lna^2 \\
2 + 1 + 1 \right 3 \, elements \right lna^3 \\
1 + 1 + 1 + 1 \right 4 \, elements \right lna^4 \\
\)



\( \mathbf{
-The \,\ integer \,\ divisor \,\ is \,\ the \,\ product \,\ of \,\ the \,\ factorials \,\ of \,\ the \,\ exponents \,\ of \,\ a_i
} \)


\(

+a.(lna.a_9 + lna^2.a_4 .a_5 + lna^2 .a_3 .a_6 + \frac{lna^3}{6} .a_3^3 + lna^2.a_2 .a_7 + lna^3.a_2 .a_3 .a_4 + \frac{lna^3}{2}.a_2^2 .a_5 + \frac{lna^4}{6} .a_2^3 .a_3 + lna^2.a_1 .a_8 + \frac{lna^3}{2} .a_1 .a_4^2 + \\
lna^3 .a_1 .a_3 .a_5 + lna^3 .a_1 .a_2 .a_6 + \frac{lna^4}{2} .a_1 .a_2 .a_3^2 + \frac{lna^4}{2} .a_1 .a_2^2 .a_4 + \frac{lna^5}{24}.a_1 .a_2^4 + \frac{lna^3}{2}.a_1^2 .a_7 + \frac{lna^4}{2}.a_1^2 .a_3 .a_4 + \frac{lna^4}{2} .a_1^2 .a_2 .a_5 + \frac{lna^5}{4}.a_1^2 .a_2^2 .a_3 + \frac{lna^4}{6} .a_1^3 .a_6 +\\
\frac{lna^5}{12} .a_1^3 .a_3^2 + \frac{lna^5}{6} .a_1^3 .a_2 .a_4 + \frac{lna^6}{36}.a_1^3 .a_2^3 + \frac{lna^5}{24}.a_1^4 .a_5 + \frac{lna^6}{24}.a_1^4 .a_2 .a_3 + \frac{lna^6}{120}.a_1^5 .a_4 + \frac{lna^7}{240} .a_1^5 .a_2^2 + \frac{lna^7}{720} .a_1^6 .a_3 + \frac{lna^8}{5040}.a_1^7 .a_2 + \frac{lna^9}{362880}.a_1^9 ).x^9
\)

\( \small {
9\,\,\right\,\, a_9^1\,\,\right\,\, 1! \,=\, 1 \\
1,\, 8\,\,\right\,\,a_1^1\,.\,a_8^1\,\,\right\,\,1!.1! \,=\, 1 \\
2,\, 7\,\,\right\,\, a_2^1\,.\,a_7^1\,\,\right\,\, 1!.1! \,=\, 1 \\
3,\, 6\,\,\right\,\, a_3^1\,.\,a_6^1\,\,\right\,\, 1!.1! \,=\, 1 \\
4,\, 5\,\,\right\,\, a_4^1\,.\,a_5^1\,\,\right\,\, 1!.1! \,=\, 1 \\
1,\, 1,\, 7\,\,\right\,\,a_1^2\,.\,a_7^1\,\,\right\,\,2!.1! \,=\, 2 \\
1,\, 2,\, 6\,\,\right\,\,a_1^1\,.\,a_2^1\,.\,a_6^1\,\,\right\,\,1!.1!.1! \,=\, 1 \\
1,\, 3,\, 5\,\,\right\,\,a_1^1\,.\,a_3^1\,.\,a_5^1\,\,\right\,\,1!.1!.1! \,=\, 1 \\
1,\, 4,\, 4\,\,\right\,\,a_1^1\,.\,a_4^2\,\,\right\,\,1!.2! \,=\, 2 \\
2,\, 2,\, 5\,\,\right\,\, a_2^2\,.\,a_5^1\,\,\right\,\, 2!.1! \,=\, 2 \\
2,\, 3,\, 4\,\,\right\,\, a_2^1\,.\,a_3^1\,.\,a_4^1\,\,\right\,\, 1!.1!.1! \,=\, 1 \\
3,\, 3,\, 3\,\,\right\,\, a_3^3\,\,\right\,\, 3! \,=\, 6 \\
1,\, 1,\, 1,\, 6\,\,\right\,\,a_1^3\,.\,a_6^1\,\,\right\,\,3!.1! \,=\, 6 \\
1,\, 1,\, 2,\, 5\,\,\right\,\,a_1^2\,.\,a_2^1\,.\,a_5^1\,\,\right\,\,2!.1!.1! \,=\, 2 \\
1,\, 1,\, 3,\, 4\,\,\right\,\,a_1^2\,.\,a_3^1\,.\,a_4^1\,\,\right\,\,2!.1!.1! \,=\, 2 \\
1,\, 2,\, 2,\, 4\,\,\right\,\,a_1^1\,.\,a_2^2\,.\,a_4^1\,\,\right\,\,1!.2!.1! \,=\, 2 \\
1,\, 2,\, 3,\, 3\,\,\right\,\,a_1^1\,.\,a_2^1\,.\,a_3^2\,\,\right\,\,1!.1!.2! \,=\, 2 \\
2,\, 2,\, 2,\, 3\,\,\right\,\, a_2^3\,.\,a_3^1\,\,\right\,\, 3!.1! \,=\, 6 \\
1,\, 1,\,1,\, 1,\, 5\,\,\right\,\,a_1^4\,.\,a_5^1\,\,\right\,\,4!.1! \,=\, 24 \\
1,\, 1,\, 1,\, 2,\, 4\,\,\right\,\,a_1^3\,.\,a_2^1\,.\,a_4^1\,\,\right\,\,3!.1!.1! \,=\, 6 \\
1,\, 1,\, 1,\, 3,\, 3\,\,\right\,\,a_1^3\,.\,a_3^2\,\,\right\,\,3!.2! \,=\, 12 \\
1,\, 1,\, 2,\, 2,\, 3\,\,\right\,\,a_1^2\,.\,a_2^2\,.\,a_3^1\,\,\right\,\,2!.2!.1! \,=\, 4 \\
1,\, 2,\, 2,\, 2,\, 2\,\,\right\,\,a_1^1\,.\,a_2^4\,\,\right\,\,1!.4! \,=\, 24 \\
1,\, 1,\, 1,\, 1,\, 1,\, 4\,\,\right\,\,a_1^5\,.\,a_4^1\,\,\right\,\,5!.1! \,=\, 120 \\
1,\, 1,\, 1,\, 1,\, 2,\, 3\,\,\right\,\,a_1^4\,.\,a_2^1\,.\,a_3^1\,\,\right\,\,4!.1!.1! \,=\, 24 \\
1,\, 1,\, 1,\, 2,\, 2,\, 2\,\,\right\,\,a_1^3\,.\,a_2^3\,\,\right\,\,3!.3! \,=\, 36 \\
1,\, 1,\, 1,\, 1,\, 1,\, 1,\, 3\,\,\right\,\,a_1^6\,.\,a_3^1\,\,\right\,\,6!.1! \,=\, 720 \\
1,\, 1,\, 1,\, 1,\, 1,\, 2,\, 2\,\,\right\,\,a_1^5\,.\,a_2^2\,\,\right\,\,5!.2! \,=\, 240 \\
1,\, 1,\, 1,\, 1,\, 1,\, 1,\, 1,\, 2\,\,\right\,\,a_1^7\,.\,a_2^1\,\,\right\,\,7!.1! \,=\, 5040 \\
1,\, 1,\, 1,\, 1,\, 1,\, 1,\, 1,\, 1,\, 1\,\,\right\,\,a_1^9\,\,\right\,\,9! \,=\, 362880 \\
} \)