We report the discovery of two unsaturated organic species, trans-(E)-cyanovinylacetylene and vinylcyanoacetylene, using the second data release of the GOTHAM deep survey towards TMC-1 with the 100 m Green Bank Telescope. For both detections, we performed velocity stacking and matched filter analyses using Markov chain Monte Carlo simulations, and for trans-(E)-cyanovinylacetylene, three rotational lines were observed at low signal-to-noise (${sim}$3$sigma$). From this analysis, we derive column densities of $2times10^{11}$ and $3times10^{11}$ cm$^{-2}$ for vinylcyanoacetylene and trans-(E)-cyanovinylacetylene, respectively, and an upper limit of $<2times10^{11}$ cm$^{-2}$ for trans-(Z)-cyanovinylacetylene. Comparisons with G3//B3LYP semi-empirical thermochemical calculations indicate abundances of the [H$_3$C$_5$N}] isomers are not consistent with their thermodynamic stability, and instead their abundances are mainly driven by dynamics. We provide discussion into how these species may be formed in TMC-1, with reference to related species like vinyl cyanide (CH$_2$=CHCN). As part of this discussion, we performed the same analysis for ethyl cyanide (CH$_3$CH$_2$CN), the hydrogenation product of CH$_2$=CHCN. This analysis provides evidence -- at 4.17$sigma$ significance -- an upper limit to the column density of $<4times10^{11}$ cm$^{-2}$; an order of magnitude lower than previous upper limits towards this source.
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