[seqfan] A347204 bits, products, and Bell numbers

[Kevin Ryde]

A347204 in draft at this moment has a definition by recurrence on
clearing of 1-bits in n.  The effect is certain products from the
positions of 1-bits in n and I wonder if the type of product is known.

  ascending 1 <= w < x < y < z
  a(n) = 1
         + w + x + y + z
         + w*(x-1) + w*(y-1) + w*(z-1) + x*(y-1) + x*(z-1) + y*(z-1)
         + w*(x-1)*(y-2) + w*(x-1)*(z-2) + w*(y-1)*(z-2) + x*(y-1)*(z-2)
         + w*(x-1)*(y-2)*(z-3)

The pattern is all combinations of w,x,y,z taken none at a time up to
all 4 at a time, and put a -0, -1, -2 etc on successive terms in each
product.

Is this some combinatorial count or algebraic operation?

If it weren't for the decrements -0,-1,-2 then it's as simple as
(1+w)*(1+x)*etc.  But with them?

A vector v=[w,x,y,z,...] of any number of terms is possible.  Mikhail
has [1,2,...,n] = Bell(n+1) and odd numbers = A002720.  Even numbers
seem to be A000262, or 3*i seems A049118.