I'm noticing a pattern in the numerators. To avoid getting indices wrong, I'll write out an example: $$(22680, 18900, 4410, 255, 1) = (36 \times 630, 16 \times 630+21 \times 420, 9 \times 420 + 10 \times 63, 4 \times 63 + 3 \times 1, 1)$$ Here $(36, 21, 10, 3, 1)$ is the odd indexed triangular numbers and $(16, 9, 4,1)$ are the squares.
More thoughts later if I have any.