[seqfan] Re: Question regarding the "inverse modulo 2 binomial transform"

[Neil Sloane]

Thomas,
Judging by many sequences like A100735 submitted by Paul Barry, it appears
that the mod 2 binomial transform and its inverse are defined by

B(n) = Sum_{k=0..n} (binomial(n,k) mod 2)*A(k),
A(n) = Sum_{k=0..n} (-1)^A010060(n-k)*(binomial(n, k) mod 2)*B(k).

For comparison, the ordinary binomial transform and its inverse are
b(n) = Sum_{k=0..n} binomial(n,k)*a(k).
a(n) = Sum_{k=0..n} (-1)^k*binomial(n, k)*b(k),
These are standard, of course - see the Transforms link at the foot of any
OEIS page, or
M. Bernstein & N. J. A. Sloane, Some canonical sequences of integers,
Linear Algebra and its Applications, 226-228 (1995), 57-72.

I haven't seen the mod 2 version used outside of the OEIS, so
if you want a reference, cite A100735 - I just added the formal definition
there.

There were a lot of major typos in this bunch of sequences, so obviously
they have not seen a lot of use in the past 15 years.
For example A100681 had the definition
Inverse modulo 2 modulo transform of 10^n.



On Wed, Dec 18, 2019 at 9:47 PM Thomas Baruchel <baruchel at gmx.com> wrote:

> Hi,
>
> I am considering writing some ideas concerning something that seems
> to match the "inverse modulo 2 binomial transform" in the OEIS. A search
> with Google gives a very tiny number of results on the OEIS, but I
> wonder whether I could find some precise reference to this transform.
>
> Is there such a reference? Could I find somewhere the exact definition
> of this transform? How could I cite such a definition?
>
> Best regards,
>
> --
> Thomas Baruchel
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>