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Trying to build an approximation, I encountered the following polynomials $$\left( \begin{array}{cc} 1 & 1 \\ 2 & x-3 \\ 3 & x^2-30 x+45 \\ 4 & x^3-273 x^2+1575 x-1575 \\ 5 & x^4-2460 x^3+43470 x^2-132300 x+99225 \\ 6 & x^5-22143 x^4+1123650 x^3-8004150 x^2+16372125 x-9823275 \\ 7 & x^6-199290 x^5+28423395 x^4-431531100 x^3+1830403575 x^2-2809456650 x+1404728325 \end{array} \right)$$

What I noticed is that

  • the constant term is $(-1)^{n}\,\frac{2^{2 n+1} \Gamma \left(n+\frac{1}{2}\right) \Gamma \left(n+\frac{3}{2}\right)}{\pi }$
  • the coefficient of the second highest power is $\frac{3}{8} \left(1-9^n\right)$
  • the coefficient of the third highest power is $\frac{15}{128} \left(2-3^{2 n+5}+5^{2 n+4}\right)$ but this does not help me much.

Any idea or suggestion would be very welcomed

Claude Leibovici
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  • Hi Claude. I computed the roots of the first polynomials : they are all real and positive and the biggest root represents a huge percentage of the trace: $97.8 %, 99.3 %, 99.7 %$ for degrees 3,4,5 resp. A (non rare) phenomena which could hopefully be compensated by considering reciprocal polynomials ( P(1/x) instead of P(x))... – Jean Marie Apr 13 '22 at 15:57
  • FWIW the constant term is A079484, which has several references listed, though not obviously related to these polynomials. – dxiv Apr 13 '22 at 16:33
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    small thing, next diagonal, namely 1575, 132300, 8004159, I get: if you add $\frac{175}{1024}$ to each term, recurrence has characteristic polynomial $(x-9)(x-25)(x-49)$ – Will Jagy Apr 13 '22 at 18:11

1 Answers1

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$\{-1575, -132300, -8004150, -431531100,...\}$:

$\dfrac{35}{1024}\bigg(5 - 3^{2 n + 8} + 5^{2 n + 7} - 7^{2 n + 6}\bigg)$

$\{1, -30, 1575, -132300, 16372125, -2809456650,...\}$:

$-(-1)^{n + 1} 2^{2 n} (n + 1) Pochhammer\bigg(\dfrac{3}{2}, n\bigg) Pochhammer\bigg(\dfrac{5}{2}, n\bigg)$

Dmitry Ezhov
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