Delta Difference Operation, Find Formula for a Sequence - then Solve for n

A culmination of work...after doing a difference operation on a sequence an (x) amount of times, what is the general term of the initial sequence? Then taking an unplanned detour, by applying what we know...find a general formula for a sequence and then solving for n.

Viewers can skip to the application: @13:46

Also, taking a detour into the hypothetical...but then writing a recipe...proof of existence?

Sources:

Gelfand, Gerver, Kirillov, Konstantinov and Kushnirenko - Sequences: 1969, problem 24