We present the discovery and follow-up observations of two CCSNe that occurred in the luminous infrared galaxy (LIRG), NGC3256. The first, SN2018ec, was discovered using the ESO HAWK-I/GRAAL adaptive optics seeing enhancer, and was classified as a Type Ic with a host galaxy extinction of $A_V=2.1^{+0.3}_{-0.1}$ mag. The second, AT2018cux, was discovered during the course of follow-up observations of SN2018ec, and is consistent with a sub-luminous Type IIP classification with an $A_V=2.1 pm 0.4$ mag of host extinction. A third CCSN, PSNJ10275082-4354034 in NGC3256, has previously been reported in 2014, and we recovered the source in late time archival HST imaging. Based on template light-curve fitting, we favour a Type IIn classification for it with modest host galaxy extinction of $A_V=0.3^{+0.4}_{-0.3}$ mag. We also extend our study with follow-up data of the recent Type IIb SN2019lqo and Type Ib SN2020fkb that occurred in the LIRG system Arp299 with host extinctions of $A_V=2.1^{+0.1}_{-0.3}$ and $A_V=0.4^{+0.1}_{-0.2}$ mag, respectively. Motivated by the above, we inspected, for the first time, a sample of 29 CCSNe located within a projected distance of 2.5 kpc from the host galaxy nuclei in a sample of 16 LIRGs. We find that, if star formation within these galaxies is modelled assuming a global starburst episode and normal IMF, there is evidence of a correlation between the starburst age and the CCSN subtype. We infer that the two subgroups of 14 H-poor (Type IIb/Ib/Ic/Ibn) and 15 H-rich (Type II/IIn) CCSNe have different underlying progenitor age distributions, with the H-poor progenitors being younger at 3$sigma$ significance. However, we do note that the available sample sizes of CCSNe and host LIRGs are so far small, and the statistical comparisons between subgroups do not take into account possible systematic or model errors related to the estimated starburst ages. (abridged)
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