(* Series from Enting, J Phys. A 27 (1994) 6987*)

cn={0,0,16,72,288,1200,5376,25480,125504,634608,3269680,17086168,90282240,481347152,2585485504,13974825960,75941188736,414593263952,2272626444528,12502223573304,68996534259040,381858968527680,2118806030647328,11783826597027256,65674579024955904,366728645195006000,2051443799934043632,11494250259278105304,64499139095733378176,362436080938852037648,2039249170926323834880,11487673072269872540904,64786142191741932873984,365754067103461706996304,2066925549185792626090544,11691314122170272566638200,66188283453887221177721568,375021938737150106426702208,2126523853550658555941372768};

pown=Table[n,{n,0,76,2}]

fnLT={0,0,1,2,9/2,12} (* by hand *)

fnLT=Join[{0},Drop[cn,1]/Drop[pown,1]^2];Take[fnLT,6]

fnHT={0,1,3/2,7/3,19/4,61/5} (* by hand *)

(* +2/pown=1/n from Log[Cosh^2]=-Log[1-Tanh^2] expansion *)

fnHT=Join[{0},Drop[fnLT,1]+2/Drop[pown,1]];Take[fnHT,6]

fHT[T_]=Log[2]+Sum[fnHT[[p]]Tanh[1/T]^pown[[p]],{p,Length[fnHT]}];

fLT[T_]=2/T+Sum[fnLT[[p]]*Exp[-2*pown[[p]]/T],{p,Length[fnLT]}];

f[T_]:=Piecewise[{{fLT[T],T<2.2},{fHT[T],T>2.3}}]

Plot[f[T],{T,0.1,4.0}]

Plot[{fLT[T],fHT[T]},{T,1.9,2.6}]