Scaling relations between supermassive black hole mass, M_BH, and host galaxy properties are a powerful instrument for studying their coevolution. A complete picture involving all of the black hole scaling relations, in which each relation is consistent with the others, is necessary to fully understand the black hole-galaxy connection. The relation between M_BH and the central light concentration of the surrounding bulge, quantified by the Sérsic index n, may be one of the simplest and strongest such relations, requiring only uncalibrated galaxy images. We have conducted a census of literature Sérsic index measurements for a sample of 54 local galaxies with directly measured M_BH values. We find a clear M_BH - n relation, despite an appreciable level of scatter due to the heterogeneity of the data. Given the current M_BH - L_sph and the L_sph - n relations, we have additionally derived the expected M_BH - n relations, which are marginally consistent at the 2 sigma level with the observed relations. Elliptical galaxies and the bulges of disc galaxies are each expected to follow two distinct bent M_BH - n relations due to the Sérsic/core-Sérsic divide. For the same central light concentration, we predict that M_BH in the Sérsic bulges of disc galaxies are an order magnitude higher than in Sérsic elliptical galaxies if they follow the same M_BH - L_sph relation. ; Comment: 12 pages, 6 figures, 5 tables, accepted for publication in MNRAS