  We obtain a cancellation-free formula, represented in terms of Schr\"oder
trees, for the antipode in the double tensor Hopf algebra introduced by
Ebrahimi-Fard and Patras. We apply the antipode formula in the context of
non-commutative probability and recover cumulant-moment formulas as well as a
new expression for Anshelevich's free Wick polynomials in terms of Schr\"oder
trees.
