What genetic hypothesis can explain these results? What crosses would you carry out to test this hypothesis?

there are 2 genes controlling the trait. C dominant is triangular, recessive is ovoid.
---------CC-----Cc------cC-----cc
CC---CCCC-CCCc-CCcC-CCcc
Cc----CcCC--CcCc-CccC--Cccc
cC----cCCC--cCCc-cCcC--cCcc
cc-----ccCC---ccCc-cccC---cccc

in generation 2 there are 16 possibilities, 15 of which have at least one C dominant, and 1 that is all c recessive. there's your 15:1 ratio for ya.

That double punnett square took me a lot of time, i really hope this helps.

By the way: start generation 1 with a CC and cc to get CC, Cc, cC, and cc. this will yeild the possibilities for the second square.