Should A358369 have offset 1?

[Ruud H.G. van Tol]

A016116:
? [ 2^(n\2) |n<-[ 0 ..30]]
%1 = [1, 1, 2, 2, 4, 4, 8, 8, 16, 16, 32, 32, 64, 64, 128, 128, 256,
256, 512, 512, 1024, ...]


? ET(v)= Vec( prod( k=1,#v, 1 / (1 - x^k+x*O(x^#v))^v[k] ) );
(notice that k is never zero)


? ET([ 2^(n\2) |n<-[ 0 ..30]])
%2 = [1, 1, 2, 4, 7, 13, 23, 41, 70, 124, 209, 361, 605, 1029, 1710,
2876, 4746, 7896, ...]
(not yet in OEIS)


A358369:
? ET([ 2^(n\2) |n<-[ 1 ..30]])
%3 = [1, 1, 3, 5, 12, 20, 43, 73, 146, 250, 475, 813, 1499, 2555, 4592,
7800, ...]

Does this mean that A358369 should have offset 1?

-- Ruud

[Peter Luschny]

This collection of strange formal expressions is not a good
basis for discussion. Please at least explain what they mean,
and don't assume that anyone is familiar with niche
mathematical programs like PARI.

I believe the confusion arises from mistaking the offset of
the input sequence with the offset of the output sequence.

Consider:

ET(v) = Vec(prod(k=1, #v, 1/(1-x^k+x*O(x^#v))^v[k]))
ET([ 2^n | n <- [1 ..29] ])
ET([ 2^n | n <- [0 ..30] ])

The first is A034899 and the second is A034691.
Note that both sequences have offset 0.

[Ruud H.G. van Tol]

On 2026-01-08 11:35, Peter Luschny wrote:
> [...]

> I believe the confusion arises from mistaking the offset of
> the input sequence with the offset of the output sequence.
>
> Consider:
>
> ET(v) = Vec(prod(k=1, #v, 1/(1-x^k+x*O(x^#v))^v[k]))
> ET([ 2^n | n <- [1 ..29] ])
> ET([ 2^n | n <- [0 ..30] ])
>
> The first is A034899 and the second is A034691.
> Note that both sequences have offset 0.

With those two sequences, the ambiguity is removed by explicitly
defining the start term for the transform, so that could be the way to
work with.
And then indeed the derived sequences can simply all have offset 0.

A358369 Euler transform of 2^floor(n/2), (A016116).
would then become
A358369 Euler transform of 2^floor(n/2), (A016116 with offset 1).
or
A358369 Euler transform of 2^floor(n/2), (A016116 [1, 2, 2, 4, ...]).
or similar.

Most cases I checked, do the transform on the whole sequence they refer
to, so IMO already reduce the ambiguity enough to leave as is.
Still it would be good to always define the start term with transforms,
as offsets of sequences tend to evolve.

My earlier idea, of using the offset of the derived sequence as a way to
define the start term of the source sequence, is indeed not practical,
already because the order in which such related sequences enter the
OEIS, is not predefined.

-- Ruud