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General Construction of the odd amplitudes, <1|Un (T, 0)|0>

Nest[(x*#) &, x, 5]

Nest[
      Integrate[Integrate[Exp[-x]*#, {x, 0, x}]*Exp[x], {x, 0, x}] &, 
      Exp[x] - 1, 1]

Integrate[
      Integrate[Exp[-x]*(Exp[x] - 1), {x, 0, x}]*Exp[x], {x, 0, 
        x}]

Nest[
      Integrate[Integrate[Exp[-x]*#, {x, 0, x}]*Exp[x], {x, 0, x}] &, 
      Exp[x] - 1, 0]

F[y_, 
        n_] := (-λ/ϵ)^(2*n + 1)*
        Limit[Nest[
            Integrate[
                Integrate[Exp[-x]*#, {x, 0, x}]*Exp[x], {x, 0, x}] &, 
            Exp[x] - 1, n], x -> y]

U[k_] := 
      Exp[-I*ϵ*T/2]*
        Sum[F[I*ϵ*T, n], {n, 0, k}]

U[0]

f0 = U[0]

    f1 = U[1]

    f2 = U[2]

    f3 = U[3]

    f4 = U[4]

Clear[λ]

    Clear[ϵ]

λ = 1/4
ϵ = 2
ω = 
      Sqrt[(ϵ/2)^2 + λ^2]
Plot[{(λ/ω*Sin[ω*T])^2, Abs[f0]^2, 
        Abs[f1]^2}, {T, 0, 50}, 
      PlotStyle -> {Hue[ .7], Hue[ .4], Hue[0.95]}, 
      PlotPoints -> 30]
Plot[{(λ/ω*Sin[ω*T])^2, Abs[f3]^2, 
        Abs[f4]^2}, {T, 0, 100}, 
      PlotStyle -> {Hue[ .7], Hue[ .4], Hue[0.95]}, 
      PlotPoints -> 30]