While black walnut's question has already been answered in principle and there has been some mention of 'free chromatic bass' piano accordions there is yet another wrinkle in the fabric of the story. I have a bandoneon (smaller than an accordion and larger than most concertinas) that has its button on both ends laid out in four rows. All the buttons have only the one note (whether pushing or pulling) but the 'inside row' of buttons is white and the next row is black and grouped in twos and threes with spaces between the groups; the next rows repeat this arrangement.

To all intents, this is a piano keyboard layout but using buttons instead of keys, but it is the only bandoneon I've seen with such a layout. The instrument was brought to Australia in about 1924 but the other bandoneons I've seen have all got a diatonic keyboard. I doubt it could have worked if keys were used instead of buttons,

Cheers, Rowan