[ALPHA]
I decided to plan the variant of Y sequence that (1,3) is expanded as (1,2,6,24,120,720,...) using the "factorial" sequence, called "Factorially Powerful Y sequence" (FPY sequence). This is not formalized yet.
First, we sneak a handful of expansions of FPY sequence before defining the rules:
(1,2,5) has the expansion (1,2,4,8,16,32,64,...) and corresponds to (1,3) in the regular Y sequence
(1,3) has the expansion (1,2,6,24,120,720,...)
(1,4) has the expansion (1,3,15,105,945,10395,135135,...)
(1,5) has the expansion (1,4,28,280,3640,58240,1164800,...)
(1,6) has the expansion (1,5,45,585,9945,208845,5221125,...)
(1,n+1) has the expansion (1, n, n(2n-1), n(2n-1)(3n-2), n(2n-1)(3n-2)(4n-3), n(2n-1)(3n-2)(4n-3)(5n-4), n(2n-1)(3n-2)(4n-3)(5n-4)(6n-5), ...)
And how can I find the expansion for (1,2,6)?