  In this article we prove that double quasi-Poisson algebras, which are
non-commutative analogues of quasi-Poisson manifolds, naturally give rise to
pre-Calabi-Yau algebras. This extends one of the main results in [11] (see also
[10]), where a relationship between pre-Calabi-Yau algebras and double Poisson
algebras was found. However, a major difference between the pre-Calabi-Yau
algebra constructed in the mentioned articles and the one constructed in this
work is that the higher multiplications indexed by even integers of the
underlying $A_{\infty}$-algebra structure of the pre-Calabi-Yau algebra
associated to a double quasi-Poisson algebra do not vanish, but are given by
nice cyclic expressions multiplied by explicitly determined coefficients
involving the Bernoulli numbers.
