[seqfan] Binomial and Inverse Binomial Transform -- terminology confusion

Wed Oct 21 22:44:10 CEST 2015

```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?