[seqfan] Re: polynomial-to-product transform

franktaw at netscape.net franktaw at netscape.net
Fri Nov 7 07:54:38 CET 2008

Why do you have step 2?  It seems simpler to just start with
C1 = A* =  1 + a(1)x + a(2)x^2 + a(3)x^3 +...

Then you're getting A* = (1+b(1)x)(1+b(2)x^2)...,
which unlike your formula has just one term of each
degree.  By first dividing by (1+x) you are really doing an
extra transform first.

The rule for "sign" and "nonn" is that the "nonn" keyword
should be used if all the terms in the database are non-
negative, even if there are later terms that are negative.
Use "sign" only if you are entering negative terms.

Franklin T. Adams-Watters

-----Original Message-----
From: Neil Fernandez <primeness at borve.org>

An integer sequence can be transformed as follows:

1) from the sequence A = {a(0), a(1),...},
     construct the series A* = 1 + a(1)x + a(2)x^2 + a(3)x^3 +...

2) divide A* by (1+x) to get C1 = (1 + b(1)x +...)
3) divide C1 by (1+b(1)x) to get C2 = (1 + b(2)x^2 +...)
4) divide C2 by (1+b(2)x^2) to get C3 = (1 + b(3)x^3 +...)
3) divide C3 by (1+b(3)x^3) to get C4 = (1 + b(4)x^4 +...)

giving A* = the product (1+x)(1+b(1)x)(1+b(2)x^2)...

from which we get the sequence B = {b(1),b(2),...}

If A is the prime sequence then B = {1,2,1,3,2,-4,2,5,4,-6,4,4,10,-36,..

If A is the Fibonacci sequence, beginning {1,2,...}, then B = {0,2,1,4,2

Neither of these were in the OEIS. When submitting the second, I got an
error message for leaving both "nonn" and "sign" unchecked, so I checked
"nonn" but I'm not sure that the sequence doesn't contain any negative

If we call B the Polynomial-to-Product transform of A, written PTP(A),
then questions arising include:

* for what sequences A in the OEIS is PTP(A) also in the OEIS?
* for what sequences A in the OEIS is the inverse transform PTP^(-1)(A)
also in the OEIS? (If A is the prime sequence, then the inverse
transform is {1,3,5,14,28,57,...}
* for what A does PTP(A)=A?


Neil Fernandez BA PhD


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