[seqfan] A347204 bits, products, and Bell numbers
Kevin Ryde
user42_kevin at yahoo.com.au
Sat Nov 6 05:23:34 CET 2021
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.
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