. 2025 Apr 5;44(5):227. doi: 10.1007/s40314-025-03139-5

Table 1.

Exact values of the probabilities for roots $η_{n}^{(2)}, {\bar{η}}_{n}^{(2)}, η_{n}^{(3)}$

	$k = 0$	$k = 1$	$k = 2$
$η_{2}^{(n)} (k)$	0	${(\frac{2}{5})}^{n}$	$1 - {(\frac{2}{5})}^{n}$
${\bar{η}}_{2}^{(n)} (k)$	$\frac{33}{28} {(\frac{3}{5})}^{n} - \frac{5}{28} {(\frac{1}{25})}^{n}$	$\frac{39}{28} {(\frac{3}{5})}^{n} - \frac{29}{18} {(\frac{2}{5})}^{n} + \frac{55}{252} {(\frac{1}{25})}^{n}$	$1 - \frac{18}{7} {(\frac{3}{5})}^{n} + \frac{29}{18} {(\frac{2}{5})}^{n} - \frac{5}{126} {(\frac{1}{25})}^{n}$
$η_{3}^{(n)} (k)$	$\frac{11}{14} {(\frac{3}{5})}^{n} - \frac{3}{14} {(\frac{1}{25})}^{n}$	$\frac{39}{42} {(\frac{3}{5})}^{n} - \frac{28}{42} {(\frac{2}{5})}^{n} + \frac{11}{42} {(\frac{1}{25})}^{n}$	$1 - \frac{12}{7} {(\frac{3}{5})}^{n} + \frac{2}{3} {(\frac{2}{5})}^{n} - \frac{1}{21} {(\frac{1}{25})}^{n}$