(* #################################################################### *)

N[%18,222]

(* above is e07 *)

N[%21,222]

(* above is e17 *)

N[%24,222]

(* above is e27 *)

N[%26,222]

(* above is e37*)

N[%31,222]

(* above is e47*)

N[%35,222]

(* above is e57 *)

N[%40,222]

(* above is e67 *)

N[%43,222]

(* above is e77 *)

N[%11,222]

4201.893694048535504399567941197898163720282551029839673302758621359293802613865084342353990828366182197858501647838745101583444238498791459413537890791940036376756402964350209854192995162327281830339752986585227970274320206661163081664`222.

(* above is the maximum of the k=4th derivative at x = 1, provided the 5-th derivative vanishes there  *)

(* Division of 4201.893694048535504... by the fourth deivative of T7, evaluated at x = 1, i.e (D[ChebyshevT[7,x],{x,4}])/.x->1 = 40320   gives the diminishing constant M7,3 = 0.1042136332849339162797 *)

RootReduce[%49]

N[%50,222]

0.1042136332849339162797511890178050139811578013648273728497707991408555010568914951473798112804654311060976810924563180828765735178199105024656135389581334334418838393592348762364631199197005774263477121276434828365643432590937788462714345964711544597`222.