  We determine the orbital eccentricities of individual small Kepler planets,
through a combination of asteroseismology and transit light-curve analysis. We
are able to constrain the eccentricities of 51 systems with a single transiting
planet, which supplement our previous measurements of 66 planets in
multi-planet systems. Through a Bayesian hierarchical analysis, we find
evidence that systems with only one detected transiting planet have a different
eccentricity distribution than systems with multiple detected transiting
planets. The eccentricity distribution of the single-transiting systems is well
described by the positive half of a zero-mean Gaussian distribution with a
dispersion $\sigma_e = 0.32 \pm 0.06$, while the multiple-transit systems are
consistent with $\sigma_e = 0.083^{+0.015}_{-0.020}$. A mixture model suggests
a fraction of $0.76^{+0.21}_{-0.12}$ of single-transiting systems have a
moderate eccentricity, represented by a Rayleigh distribution that peaks at
$0.26^{+0.04}_{-0.06}$. This finding may reflect differences in the formation
pathways of systems with different numbers of transiting planets. We
investigate the possibility that eccentricities are "self-excited" in closely
packed planetary systems, as well as the influence of long-period giant
companion planets. We find that both mechanisms can qualitatively explain the
observations. We do not find any evidence for a correlation between
eccentricity and stellar metallicity, as has been seen for giant planets.
Neither do we find any evidence that orbital eccentricity is linked to the
detection of a companion star. Along with this paper we make available all of
the parameters and uncertainties in the eccentricity distributions, as well as
the properties of individual systems, for use in future studies.
