Neil:

>  3.
>  # BIN1 was introduced by Don Zagier (see M. Kaneko,
>  # "A recurrence formula for the Bernoulli numbers",
>  # Proc. Japan Acad., 71 A (1995), 192-193). 

>  # It is an involution on the class of sequences a = [a_0, a_1, a_2, ...],
>  # sending a to b where b_n = (-1)^n Sum_{i=0..n} binomial(n+1,i+1) a_i.

>  BIN1:=proc(a)  local b,i,j,k:
>  if whattype(a) <> list then RETURN([]); fi:
>  b:=[]:
>  for i to nops(a) do j:=i-1; b:=[op(b), (-1)^j*add( binomial(j+1,k+1)*a[k+1] ,
>          k=0..j)]: od:
>  RETURN(b);
>  end:

The sequence that shifts left under BIN1 is
1 1 -3 3 -1 -1 1 1 -3 3 -1 -1 1 1 -3 3 -1 -1 1 1...

a repeating sequence. (No I'm not submitting it.)

However, the sequence that shifts left twice under BIN1
1 1 1 -3 7 -11 17 -27 45 -75 123 -199 321 -519 841...
looks more interesting.

The definition suggests a generalization to BINk with binomial(n+k,i+k)
in the definition. I noticed that BIN2 at least is also involutional.

>  (The signs could be omitted, but Zagier needed them for his application, so
>  i have left them in.)

The unsigned transform applied to 2^n gives (3^(n+1)-1)/2 A003462(n+1).

A040027(n+1) shifts left under the unsigned transform.

>  4.
>  # The "Stirling-Bernoulli transform" maps a sequence b_0, b_1, b_2, ...
>  # to a sequence c_0, c_1, c_2, ..., where if B has o.g.f. B(x), c has
>  # e.g.f. exp(x)*B(1-exp(x)).  [Yes, the ogf becomes an egf.]
>  # More explicitly,
>  # c_n = Sum_{m=0..n} (-1)^m*m!*Stirling2(n+1,m+1)*b_m.

This can be described as:
Multiply each term by n!
Perform alternating sign Stirling transform
Perform Binomial transform

The sequence that shifts left:
1 1 0 -2 6 250 -27090 -20110502 100802987166...
grows fast.

Christian