Here is a version that produces exact results and memoizes the derivatives and polynomials for efficiency:
airyZeta`dAA[n_ /; IntegerQ[n] && n >= 0] :=
airyZeta`dAA[n] =
If[n == 0, AiryAi[airyZeta`z]^2,
D[airyZeta`dAA[n - 1], airyZeta`z]]
airyZeta`dAB[n_ /; IntegerQ[n] && n >= 0] :=
airyZeta`dAB[n] =
If[n == 0, AiryAi[airyZeta`z] AiryBi[airyZeta`z],
D[airyZeta`dAB[n - 1], airyZeta`z]]
airyZeta`dBoA[n_ /; IntegerQ[n] && n >= 0] :=
airyZeta`dBoA[n] =
If[n == 0, AiryBi[airyZeta`z]/AiryAi[airyZeta`z],
D[airyZeta`dBoA[n - 1], airyZeta`z]]
airyZeta`poly[n_ /; IntegerQ[n] && n >= 2] :=
airyZeta`poly[
n] = \[Pi]π/
Gamma[n] (airyZeta`dBoA[n - 1]/(3^(2/3) Gamma[1/3]^2) +
airyZeta`dAB[n - 2] -
Sum[Binomial[n - 1, j] airyZeta`dBoA[n - 1 - j] airyZeta`dAA[
j - 1], {j, 1, n - 1}]) /. airyZeta`z -> 0 /.
Gamma[2/3] -> (2 \[Pi]π)/(Sqrt[3] Gamma[1/3]) /.
Gamma[1/3] -> Sqrt[(2 \[Pi]π)/(3^(1/6) airyZeta`x)] // Expand
airyZeta[n_ /; IntegerQ[n] && n >= 2] :=
airyZeta[n] =
airyZeta`poly[n] /. airyZeta`x -> (2 \[Pi]π)/(3^(1/6) Gamma[1/3]^2)
airyZetaN[n_ /; IntegerQ[n] && n >= 2] :=
airyZetaN[n] =
airyZeta`poly[n] /.
airyZeta`x -> N[(2 \[Pi]π)/(3^(1/6) Gamma[1/3]^2)]
It generates the Airy Zeta polynomials, e.g.:
$$ \begin{array}{c} x^2 \\ x^3-\frac{1}{2} \\ x^4-\frac{x}{3} \\ x^5-\frac{5 x^2}{12} \\ x^6-\frac{x^3}{2}+\frac{1}{20} \\ x^7-\frac{7 x^4}{12}+\frac{13 x}{180} \\ x^8-\frac{2 x^5}{3}+\frac{139 x^2}{1260} \\ x^9-\frac{3 x^6}{4}+\frac{87 x^3}{560}-\frac{1}{160} \\ x^{10}-\frac{5 x^7}{6}+\frac{209 x^4}{1008}-\frac{17 x}{1296} \\ x^{11}-\frac{11 x^8}{12}+\frac{671 x^5}{2520}-\frac{2167 x^2}{90720} \\ x^{12}-x^9+\frac{93 x^6}{280}-\frac{13 x^3}{336}+\frac{7}{8800} \\ x^{13}-\frac{13 x^{10}}{12}+\frac{2041 x^7}{5040}-\frac{10543 x^4}{181440}+\frac{9301 x}{4276800} \\ x^{14}-\frac{7 x^{11}}{6}+\frac{349 x^8}{720}-\frac{67 x^5}{810}+\frac{1797097 x^2}{389188800} \\ x^{15}-\frac{5 x^{12}}{4}+\frac{4 x^9}{7}-\frac{19 x^6}{168}+\frac{56909 x^3}{6726720}-\frac{1}{9856} \\ x^{16}-\frac{4 x^{13}}{3}+\frac{419 x^{10}}{630}-\frac{1699 x^7}{11340}+\frac{7688249 x^4}{544864320}-\frac{30671 x}{89812800} \\ x^{17}-\frac{17 x^{14}}{12}+\frac{3859 x^{11}}{5040}-\frac{1003 x^8}{5184}+\frac{240005881 x^5}{10897286400}-\frac{620143 x^2}{747242496} \\ x^{18}-\frac{3 x^{15}}{2}+\frac{489 x^{12}}{560}-\frac{137 x^9}{560}+\frac{1466711 x^6}{44844800}-\frac{30439 x^3}{17937920}+\frac{2169}{167552000} \\ x^{19}-\frac{19 x^{16}}{12}+\frac{2489 x^{13}}{2520}-\frac{27569 x^{10}}{90720}+\frac{509000671 x^7}{10897286400}-\frac{406671269 x^4}{130767436800}+\frac{56881351 x}{1099308672000} \\ x^{20}-\frac{5 x^{17}}{3}+\frac{559 x^{14}}{504}-\frac{3373 x^{11}}{9072}+\frac{20123489 x^8}{311351040}-\frac{17248789 x^5}{3269185920}+\frac{84600899 x^2}{596767564800} \\ \end{array} $$$$\begin{array}{c} x^2 \\ x^3-\frac{1}{2} \\ x^4-\frac{x}{3} \\ x^5-\frac{5 x^2}{12} \\ x^6-\frac{x^3}{2}+\frac{1}{20} \\ x^7-\frac{7 x^4}{12}+\frac{13 x}{180} \\ x^8-\frac{2 x^5}{3}+\frac{139 x^2}{1260} \\ x^9-\frac{3 x^6}{4}+\frac{87 x^3}{560}-\frac{1}{160} \\ x^{10}-\frac{5 x^7}{6}+\frac{209 x^4}{1008}-\frac{17 x}{1296} \\ x^{11}-\frac{11 x^8}{12}+\frac{671 x^5}{2520}-\frac{2167 x^2}{90720} \\ x^{12}-x^9+\frac{93 x^6}{280}-\frac{13 x^3}{336}+\frac{7}{8800} \\ x^{13}-\frac{13 x^{10}}{12}+\frac{2041 x^7}{5040}-\frac{10543 x^4}{181440}+\frac{9301 x}{4276800} \\ x^{14}-\frac{7 x^{11}}{6}+\frac{349 x^8}{720}-\frac{67 x^5}{810}+\frac{1797097 x^2}{389188800} \\ x^{15}-\frac{5 x^{12}}{4}+\frac{4 x^9}{7}-\frac{19 x^6}{168}+\frac{56909 x^3}{6726720}-\frac{1}{9856} \\ x^{16}-\frac{4 x^{13}}{3}+\frac{419 x^{10}}{630}-\frac{1699 x^7}{11340}+\frac{7688249 x^4}{544864320}-\frac{30671 x}{89812800} \\ x^{17}-\frac{17 x^{14}}{12}+\frac{3859 x^{11}}{5040}-\frac{1003 x^8}{5184}+\frac{240005881 x^5}{10897286400}-\frac{620143 x^2}{747242496} \\ x^{18}-\frac{3 x^{15}}{2}+\frac{489 x^{12}}{560}-\frac{137 x^9}{560}+\frac{1466711 x^6}{44844800}-\frac{30439 x^3}{17937920}+\frac{2169}{167552000} \\ x^{19}-\frac{19 x^{16}}{12}+\frac{2489 x^{13}}{2520}-\frac{27569 x^{10}}{90720}+\frac{509000671 x^7}{10897286400}-\frac{406671269 x^4}{130767436800}+\frac{56881351 x}{1099308672000} \\ x^{20}-\frac{5 x^{17}}{3}+\frac{559 x^{14}}{504}-\frac{3373 x^{11}}{9072}+\frac{20123489 x^8}{311351040}-\frac{17248789 x^5}{3269185920}+\frac{84600899 x^2}{596767564800} \\ \end{array} $$
There might be a simpler recurrence relation to generate those polynomials. This is left as an exercise for the reader. :-)
