Distribution with the following moment generating function?

Question

Can anybody suggest a distribution whose Laplace transform is the following?

$\mathrm{E}[e^{tX}] = \exp(\lambda(e^{2t}-1))$.

Note: The MGF of Poisson distribution is $\exp(\lambda(e^{t}-1))$.

Answer 1

If $Y$ is Poisson-$\lambda$ and $X=2Y$ then: $$\mathbb E(e^{tX})=\mathbb E(e^{2tY})=e^{\lambda(e^{2t}-1)}$$