# [seqfan] Re: Binomial and Inverse Binomial Transform -- terminology confusion

eli Mcfly elijaffe173 at gmail.com
Thu Oct 22 13:48:58 CEST 2015

```Hello Vladimir,
The Inverse Binomial Transform is repeatedly adding the terms in a
sequence, and the binomial transform is repeatedly finding the differences
between the terms in the sequence.
In the graphical sense, when looking at Aitken's array,  the original
function you are transforming is the diagonal and the column you create off
to the side is it's inverse.
Formula wise, the inverse transform is Sigma[ Binomial(n,k)*a(n) ] and the
transformation is Sigma[ Binomial(n,k)*a(n)*(-1)^n] .
I too went through a great deal of confusion at first, as different
websites had different definitions. I ended up following the formulas
provided by Wikipedia to understand which was which, as the definitions use
love to see a clearer definition included on OEIS as well.

Elias Jaffe
On Oct 22, 2015 12:55 AM, "Vladimir Reshetnikov" <v.reshetnikov at gmail.com>
wrote:

> Dear SeqFans,
>
> I found myself in some confusion about the Binomial Transform and the
> Inverse Binomial Transform -- which one is which?
>
> Looking at some sequences in OEIS, e.g.
> http://oeis.org/A007317 Binomial transform of Catalan numbers.
> http://oeis.org/A081567 Second binomial transform of F(n+1).
> http://oeis.org/A034943 Binomial transform of Padovan sequence A000931.
> http://oeis.org/A033321 Binomial transform of Fine's sequence A000957:
> 1,0,1,2,6,18,57,186,...
> http://oeis.org/A005021 Random walks (binomial transform of A006054).
> http://oeis.org/A126930 Inverse binomial transform of A005043.
> http://oeis.org/A084102 Inverse binomial transform of A084101.
>
> it seems that they define the Binomial Transform of a(n) as
> b(n)=sum(k=0..n, binomial(n,k)*a(n)), and the Inverse Binomial
> Transform of a(n) as b(n)=sum(k=0..n, (-1)^(n+k)*binomial(n,k)*a(n)).
>
> Recently I also made a submission following this convention:
> http://oeis.org/A263529 Binomial transform of double factorial n!!
> (A006882).
>
> I was about to submit a new sequence -- Inverse binomial transform of
> double factorial n!! -- but I looked at other sources, and they used
> different definitions.
>
> http://mathworld.wolfram.com/BinomialTransform.html
> says that the Binomial transform of a(n) is b(n)=sum(k=0..n,
> (-1)^(n-k)*binomial(n,k)*a(n)).
>
> http://en.wikipedia.org/wiki/Binomial_transform
> says that the Binomial transform of a(n) is b(n)=sum(k=0..n,
> (-1)^k*binomial(n,k)*a(n)).
>
> So now I'm a bit confused. Are the definitions used in OEIS commonly
> accepted? Are they applied uniformly across all sequences in OEIS?
> they be explicitly included in the OEIS stylesheet
> (http://oeis.org/wiki/Style_Sheet)?
>
> --
> Thanks