No, it doesn't exist. In general, the Laplace transform of tntn is Γ(n+1)sn+1Γ(n+1)sn+1, and Γ(n)Γ(n) isn't defined on 0,−1,−2,−3...0,−1,−2,−3... This integral is the definition of the Laplace transforms, so the transform doesn't exist if the integral doesn't. While there are other integral transforms that could transform 1t1t in a useful way, anything other than what you gave wouldn't be considered a Laplace transform anymore.Ekeeda - Electrical and Electronics Engineering
