Talk:More Transformations of Integer Sequences

Would some Administrator please edit the protected page More Transformations of Integer Sequences and remove {{Edit request}} here.

Links to sequences are broken. — Daniel Forgues 01:42, 10 February 2013 (UTC)

Errata and suggestions

These apply both to the wiki page and to http://oeis.org/transform2.html

LPAL needs the special case LPAL₁ = IDENTITY. It could alternatively be written as
B(x) = (A(x)+A(x^2)) / (1-A(x^2))

CHK_k is listed as
b_n = (MÖBIUS · AIK)_ka_n / n
but it should be
b_n = (MÖBIUS · AIK)_ka_n / k

CIK_k can be given directly using Pólya enumeration on the cyclic group as
b_n = sum{d|k and d|n} phi(d) (AIK_k/da)_n/d / k
This not only shows how to skip CHK in calculating CIK but also makes explicit the calculation of CIK₂, which is referred to by CPAL₂. Otherwise we're left to derive it independently or guess that if CIK = MÖBIUS^-1 · CHK then maybe CIK₂ = MÖBIUS^-1 · CHK₂.

Which brings me on to CPAL. The component J of the case n,k is given as
J=sum{i=1 to n/2}(AIK_k/2-1a)_(n-2i)/2
which accounts for only n-2i items and k-2 boxes. It should be
J=sum{i=1 to n/2} a_i (AIK_k/2-1a)_(n-2i)/2

The explanation could also be clearer: what exactly is the joining operation of "boxes joined"?

However, working backwards from DIK_k = (CIK_k + CPAL_k) / 2 and comparing the well-known cycle indices of the cyclic group and the dihedral group, it seems that a simpler rephrasing could be given:

CPAL_k
If n odd, k even:
b_n = sum{i>0 and i<n/2} (AIK₂a)_n-2i × (AIK_(k-2)/2a)_i / 2
If n,k even:
b_n = (AIK_k/2a)_n/2 / 2 + sum{i>0 and i<n/2} (AIK₂a)_n-2i × (AIK_(k-2)/2a)_i / 2
If k odd: b_n = sum{i>0 and i<n/2} a_n-2i × (AIK_(k-1)/2a)_i

Or even simpler:

CPAL
2B(x) = (1+A(x))² / (1 - A(x²)) - 1

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It might also be nice to include LPAL and CPAL in the catalogue.

LPAL sequences

A057427	LPAL s1
A060546	LPAL s2
A056449	LPAL s3
A056450	LPAL s4
A056451	LPAL s5
A163403	LPAL all1
A068911	(LPAL all2) + s1n+1
A003945	(LPAL all3) + s1n+1
A000045	(LPAL codd)2n + s1n+1
A000045	(LPAL codd)2n-3
A000045	(LPAL noone)2n-2 + s1n
A000045	(LPAL noone)2n+1
A000045	(LPAL twoone)n-3 + s1n+1

CPAL sequences

A057427	CPAL s1
A029744	CPAL s2
A182751	CPAL s3
A056486	CPAL s4
A052955	(CPAL all1)n+1
A068156	(CPAL all3) + s1n+1
A000071	(CPAL codd)2n-4 - s1n-1
A000045	(CPAL codd)2n-3
A000071	(CPAL noone)2n-4 - s1n-1
A000071	(CPAL noone)2n-1
A000071	(CPAL twoone)n-3

Independent checking of the offsets is welcomed.

Peter J. Taylor 21:24, 22 September 2017 (UTC)