  Context. The 3.3 $\mu$m aromatic C-H stretching band of polycyclic aromatic
hydrocarbon (PAH) molecules seen in a wide variety of astrophysical regions is
often accompanied by a series of weak satellite bands at ~3.4-3.6 $\mu$m. One
of these sources, IRAS 21282+5050, a planetary nebula, also exhibits a weak
band at ~1.68 $\mu$m. While the satellite features at ~3.4-3.6 $\mu$m are often
attributed to the anharmonicities of PAHs, it is not clear whether overtones or
combination bands dominate the 1.68 $\mu$m feature.
  Aims. In this work, we examine the anharmonic spectra of eight PAH molecules,
including anthracene, tetracene, pentacene, phenanthrene, chrysene,
benz[a]anthracene, pyrene, and perylene, to explore the origin of the infrared
bands in the 1.6-1.7 $\mu$m waveelngth region.
  Methods. Density Functional Theory (DFT) in combination with the vibrational
second-order perturbation theory (VPT2) is utilized for computing the
anharmonic spectra of PAHs. To simulate the vibrational excitation process of
PAHs, the Wang-Landau random walk technique is employed.
  Results. All the dominant bands in the 1.6-1.7 $\mu$m wavelength range and in
the 3.1-3.5 $\mu$m C-H stretching region are calculated and tabulated. It is
demonstrated that combination bands dominate the 1.6-1.7 $\mu$m region, while
overtones are rare and weak in this region. We also calculate the intensity
ratios of the 3.1-3.5 $\mu$m C-H stretching features to the bands in the
1.6-1.7 $\mu$m region, $I_{3.1-3.5}/I_{1.6-1.7}$, for both ground and
vibrationally excited states. On average, we obtain $\langle
I_{3.1-3.5}/I_{1.6-1.7} \rangle$ $\approx$ 12.6 and $\langle
I_{3.1-3.5}/I_{1.6-1.7} \rangle$ $\approx$ 17.6 for PAHs at ground states and
at vibrationally excited states, respectively.
