[New transforms]

[Christian G.Bower]

"N. J. A. Sloane" <njas at research.att.com> wrote:
> Dear SeqFans

> In lieu of new year's greetings, this message describes
> four transformations of sequences that have come up recently.
[...]
> 2. Take the sum of 2 or more consecutive terms of a sequence.
[...]
This can also be described as differences of the sequence of partial
sums. E.g.

If b is the transform of a, then let c be the sequence of partial sums
of a, b consists of all numbers that can be written as c(i)-c(j) for
some j>i+1. So the set of numbers the sum of 2 or more consective odds
would be differences of nonconsective squares.

> 3.
> # BIN1 was introduced by Don Zagier (see M. Kaneko,
> # "A recurrence formula for the Bernoulli numbers",
> # Proc. Japan Acad., 71 A (1995), 192-193). 

> # It is an involution on the class of sequences a = [a_0, a_1, a_2, ...],
> # sending a to b where b_n = (-1)^n Sum_{i=0..n} binomial(n+1,i+1) a_i.
[...]
So this is just to take the sequence b, shift right, apply the binomial
transform and alternate signs.


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