WHAT IS GOING ON HERE!?

[John Layman]

Antti Karttunen, Dec 11, 2003:  "So, what is going on here!?", asked in
reference to INVERT transforms of various sequences reported by Christian
Bower, myself, and Superseeker.

ME: That is just what I have been wondering since I saw Christian's results and 
checked several sequence entries located by searching for entries containing
the word "invert".

First, let me say that my INV implementation of the INVERT transform ALWAYS
assumes an offset of 1, just as if one had stumbled upon a sequence of integers
with no consideration of their meaning.  Although I don't do Maple, this is
what I believe Neil's routine INVERT in TRANSFORMS.TXT does, since the sequence
is represented by a 'list':

  # Cameron's A and inverse:  1+SUM a_n x^n = 1/(1 - SUM b_n x^n); n=1..inf
  # i.e. 1 + A(x) = 1/(1 - B(x) )
  # a's from b's:  
  INVERT:=proc(a)  local t1,t2,x,b,n:
  if whattype(a) <> list then RETURN([]); fi:
  n:=nops(a)+1:
  b:=listtoseries(a,x,'ogf'):
  t1:=series(1/(1-x*b),x,n):
  t2:=subsop(1=NULL, seriestolist(t1,'ogf')):
  if type(t2,list(integer)) then RETURN(t2) else RETURN([]): fi:
  end:


Here are some additional examples of inconsistent results:

(1) Christian gets (INVERT){0,1,1,2,3,5,8,13,21,34,...}
	={0,1,2,5,12,29,70,...}=A000129, whereas
    I get (INV){0,1,1,2,3,5,8,13,21,34,...}
	={0,1,1,3,5,11,21,43,...} = A001045
    and (INV){1,1,2,3,5,8,13,...}={1,2,5,12,29,70,...}, which is
	A000129 with the first term, 0, deleted.

(2) The OEIS states that (INVERT)A000108=(INVERT){1,1,2,5,14,42,132,...}
	={1,1,3,7,20,59,184,...}=A030238, whereas
    I get (INV){1,1,2,5,14,42,132,429,...}={1,2,5,14,42,132,149},
    i.e., (INV) shifts A000108 to the left by one term;
    and (INV){1,2,5,14,42,132,...}={1,3,10,35,126,462,...}=A001700.
 
(3) The OEIS states (INVERT)A000290=(INVERT){0,1,4,9,16,25,36,...}
	={1,5,18,63,221,776,...}=A033453, whereas
    I get (INV)A000290={0,1,4,10,24,60,152,384,...} which is NOT IN THE OEIS,
    and (INV){1,4,9,25,36,49,...}=A033453.

(4) The OEIS states that (INVERT^3) shifts A007564 left one term, 
    in COMPLETE AGREEMENT with my (INV):
    (INV^3){1,1,4,19,100,562,3304,20071,124996,...}
          ={1,4,19,100,562,3304,20071,124996,..}

(5) The OEIS states (INVERT)A002426=(INVERT){1,1,3,7,19,51,141,393,...}
     =A007971={0,1,2,2,4,8,18,42,...}, whereas
    I get (INV){1,1,3,7,19,51,141,393,...}={1,2,6,18,56,176,558,...}, which is
    NOT in the OEIS.

...and I feel certain that other discrepancies could be found.

John