  Three formaldehyde lines were observed (H$_2$CO 3$_{03}$--2$_{02}$, H$_2$CO
3$_{22}$--2$_{21}$, and H$_2$CO 3$_{21}$--2$_{20}$) in the protoplanetary disk
around the Herbig Ae star HD 163296 with ALMA at 0.5 arcsecond (60 AU) spatial
resolution. H$_2$CO 3$_{03}$--2$_{02}$ was readily detected via imaging, while
the weaker H$_2$CO 3$_{22}$--2$_{21}$ and H$_2$CO 3$_{21}$--2$_{20}$ lines
required matched filter analysis to detect. H$_2$CO is present throughout most
of the gaseous disk, extending out to 550 AU. An apparent 50 AU inner radius of
the H$_2$CO emission is likely caused by an optically thick dust continuum. The
H$_2$CO radial intensity profile shows a peak at 100 AU and a secondary bump at
around 300 AU, suggesting increased production in the outer disk. Different
parameterizations of the H$_2$CO abundance were compared to the observed
visibilities with $\chi^2$ minimization, using either a characteristic
temperature, a characteristic radius or a radial power law index to describe
the H$_2$CO chemistry. Similar models were applied to ALMA Science Verification
data of C$^{18}$O. In all modeling scenarios, fits to the H$_2$CO data show an
increased abundance in the outer disk. The overall best-fit H$_2$CO model shows
a factor of two enhancement beyond a radius of 270$\pm$20 AU, with an inner
abundance of $2\!-\!5 \times 10^{-12}$. The H$_2$CO emitting region has a lower
limit on the kinetic temperature of $T > 20$ K. The C$^{18}$O modeling suggests
an order of magnitude depletion in the outer disk and an abundance of $4\!-\!12
\times 10^{-8}$ in the inner disk. The increase in H$_2$CO outer disk emission
could be a result of hydrogenation of CO ices on dust grains that are then
sublimated via thermal desorption or UV photodesorption, or more efficient
gas-phase production beyond about 300 AU if CO is photodisocciated in this
region.
