The functionality outlined below, and much more, is implemented in Hodge diamond cutter, which can be used in Sage. If you use it for your research, please cite it using DOI.

I could keep it for the fortnightly links, but the Hodge diamond lepidopterist in me is too excited. The Hodge numbers for O'Grady's sporadic 10-dimensional example have been computed, by Mark Andrea de Cataldo, Antonio Rapagnetta and Giulia Saccà in their paper The Hodge numbers of O'Grady 10 via Ngô strings.

I have nothing useful to say about the proof, as it is far outside my comfort zone. But I did add it to my collection of butterflies Hodge diamonds. In its full glory it looks like:

$$ \begin{array}{ccccccccccccccccccccc} & & & & & & & & & 1 & & & & & & & & & \\ & & & & & & & & 0 & & 0 & & & & & & & & \\ & & & & & & & 1 & & 22 & & 1 & & & & & & & & \\ & & & & & & 0 & & 0 & & 0 & & 0 & & & & & & \\ & & & & & 1 & & 22 & & 254 & & 22 & & 1 & & & & & & \\ & & & & 0 & & 0 & & 0 & & 0 & & 0 & & 0 & & & & & \\ & & & & 1 & & 23 & & 276 & & 2299 & & 276 & & 23 & & 1 & & & & \\ & & & 0 & & 0 & & 0 & & 0 & & 0 & & 0 & & 0 & & 0 & & & \\ & & 1 & & 22 & & 276 & & 2531 & & 16490 & & 2531 & & 276 & & 22 & & 1 & & \\ & 0 & & 0 & & 0 & & 0 & & 0 & & 0 & & 0 & & 0 & & 0 & \\ 1 & & 22 & & 254 & & 2299 & & 16490 & & 88024 & & 16490 & & 2299 & & 254 & & 22 & & 1 & \\ & 0 & & 0 & & 0 & & 0 & & 0 & & 0 & & 0 & & 0 & & 0 & \\ & & 1 & & 22 & & 276 & & 2531 & & 16490 & & 2531 & & 276 & & 22 & & 1 & & \\ & & & 0 & & 0 & & 0 & & 0 & & 0 & & 0 & & 0 & & & \\ & & & & 1 & & 23 & & 276 & & 2299 & & 276 & & 23 & & 1 & & & & \\ & & & & 0 & & 0 & & 0 & & 0 & & 0 & & 0 & & & & & \\ & & & & & 1 & & 22 & & 254 & & 22 & & 1 & & & & & & \\ & & & & & & 0 & & 0 & & 0 & & 0 & & & & & & \\ & & & & & & & 1 & & 22 & & 1 & & & & & & & & \\ & & & & & & & & 0 & & 0 & & & & & & & & \\ & & & & & & & & & 1 & & & & & & & & & \\ \end{array} $$