  In this paper we investigate dark matter structure formation in the normal
branch of the Dvali-Gabadadze-Porrati (nDGP) model using the PINOCCHIO
algorithm. We first present 2nd order Lagrangian perturbation theory for the
nDGP model, which shows that the 1st- and 2nd-order growth functions in nDGP
are larger than those in {\Lambda}CDM. We then examine the dynamics of
ellipsoidal collapse in nDGP, which is accelerated compared to {\Lambda}CDM due
to enhanced gravitational interactions. Running the nDGP-PINOCCHIO code with a
box size of 512 Mpc/h and 1024*1024*1024 particles, we analyze the statistical
properties of the output halo catalogs, including the halo power spectrum and
halo mass function. The calibrated PINOCCHIO halo power spectrum agrees with
N-body simulations within 5% in the comoving wavenumber range k < 0.3 (h/Mpc)
at redshift z = 0. The agreement is extended to smaller scales for higher
redshifts. For the cumulative halo mass function, the agreement between N-body
and PINOCCHIO is also within the simulation scatter.
