  We investigate the structural aspects of genuine multipartite entanglement in
Gaussian states of continuous variable systems. Generalizing the results of
[Adesso & Illuminati, Phys. Rev. Lett. 99, 150501 (2007)], we analyze whether
the entanglement shared by blocks of modes distributes according to a strong
monogamy law. This property, once established, allows to quantify genuine
N-partite entanglement in terms of the "residual contangle" not encoded into
2,...,K,...,(N-1)-partite quantum correlations. The explicit expression of this
entanglement measure is derived, by a recursive formula, for a subclass of
Gaussian states. These are fully symmetric (permutation-invariant) states
multi-partitioned into blocks, each consisting of an arbitrary number of modes.
We compute the genuine multipartite entanglement shared by the blocks
("molecules") and investigate its scaling properties with the number and size
of the molecules, the total number of modes, the global mixedness of the state,
and the squeezed resources. To achieve the exact computation we prove a general
result of symplectic analysis: Correlations among K molecules in N-mode
multi-symmetric Gaussian states can be transformed by a local unitary operation
into correlations shared by K single modes, one per molecule, in effective
non-symmetric states. Due to this theorem, the above results extend to the
subclass of non-symmetric Gaussian states that are obtained by unitarily
localizing the multipartite molecular entanglement of symmetric states. These
findings provide evidence that distributed Gaussian entanglement is strongly
monogamous even beyond specific symmetry constraints, and that the residual
contangle is a bona fide measure of genuine multipartite entanglement for
permutation-invariant Gaussian states under any multi-partition of the modes.
