[Witt vectors A006177 and A006173]

[Antti Karttunen]

One furher note:

A006177: x,1,1,3,8,25,72,245,772,2692,8925,32065,109890,400023,1402723,5165327,18484746,68635477,248339122,930138521,3406231198,12810761323,47306348881,178987624513,665627041157,2528210175630,9456885664122

A022553:
1,1,1,3,8,25,75,245,800,2700,9225,32065,112632,400023,1432613,5170575,18783360,68635477,252085716,930138521,3446158600,12815663595,47820414961,178987624513,671825020128,2528212128750,9536894664375

  1,2,3,4,5,    7,                11,          13,                             17,                19,                                          23, ...

i.e. all the terms in prime positions (in addition to the fourth term 8)
seem to be same. But maybe this is an obvious consequence of the
definitions of Inverse Somos and Inverse Euler transforms.
(I'm abstaining from now on from coffee, thus being unable to think.)

But this reminds me also of the similar identity between two
"necklacish" sequences I found over a year ago:

A061860[p] = A061417[p] = (p-1)!+(p-1) for all prime p's.

where the former is:

ID Number: A061860
Sequence:  1,2,4,11,28,152,726,5268,40438,365944,3628810,39974466,
           479001612,6228256404,87178339984,1307706805928,
           20922789888016,355688409760972,6402373705728018,
           121645133931170028,2432902008232456692
Name:      Variant of A061417.
Comments:  Does this count some variety of necklaces?
Formula:   a(n) = (1/n)*Sum_{d|n} phi(n/d)*C(n,d)*(d!)
Maple:     [seq(A061860(j),j=1..40)]; with(numtheory); A061860 := proc(n)
              local d,s; s := 0; for d in divisors(n) do s := s +
              phi(n/d)*(binomial(n,d))*(d!); od; RETURN(s/n); end;
See also:  A061860[p] = A061417[p] = (p-1)!+(p-1) for all prime p's.
Keywords:  nonn
Offset:    0
Author(s): Antti.Karttunen at iki.fi May 11 2001


-- Antti


"Christian G.Bower" wrote:
> 
>   --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
> Antti:
> 
> > How does these transforms, somos and invsomos differ from
> > the transforms EULER and EULERi
> 
> I guess I missed the original discussion of the Somos transform. If someone can
> give me the month and year so I can dig it out of the log, that would be nice.
> 
> Anyway, the Somos transform as Wouter defined it is the same as one of a group
> of transforms I've informally called "smallest to largest." (More on that
> later.)
> 
> The Euler and Somos transforms are similar.
> 
> Euler transforms the sequence a[n] to the sequence whose generating function is:
> 
> PRODUCT (1/(1-x^i)^a[i]) i=1 to n
> 
> Somos transforms the sequence a[n] to the sequence whose generating function is:
> 
> PRODUCT (1/(1-a[i]*x^i)) i=1 to n
> 
> I don't know of too many examples of the Somos transform in the EIS, but a good
> one is A006906 which is the Somos transform of the natural numbers A000027
> 
> %I A006906 M2575
> %S A006906 1,1,3,6,14,25,56,97,198,354,672,1170,2207,3762,6786,11675,20524,34636,
> %T A006906 60258,100580,171894,285820,480497,791316,1321346,2156830,3557353,
> %U A006906 5783660,9452658,15250216,24771526,39713788,64011924,102199026
> %N A006906 Sum of products of terms in all partitions of n.
> %D A006906 G. Labelle, personal communication.
> %F A006906 G.f.: 1 / Product (1-kx^k).
> %e A006906 The partitions of 4 are 4, 1+3, 2+2, 2+1+1, 1+1+1+1, the corresponding products are 4,3,4,2,1, and their sum is a(4) = 14.
> %Y A006906 Cf. A007870.
> %K A006906 nonn,nice,easy
> %O A006906 0,3
> %A A006906 Simon Plouffe (plouffe at math.uqam.ca)
> %E A006906 More terms from Vladeta Jovovic (vladeta at Eunet.yu), Oct 04 2001
> 
> Both Somos and Euler transforms applied to the all 1's sequence give the
> partition numbers A000041.
> 
> A070933 is the Somos transform of the all 2's sequence.
> 
> I used the name smallest to largest because the transform can be described as
> follows:
> 
> Let sequence a describe the number of some type of unlabeled structures that
> can be built from n points.
> 
> The Euler transform of a gives the number of "sets" of those structures
> consisting of n points. (i.e. if n=3 you have a 3-structure by itself, or a
> 2-structure and a 1-structure or 3 1-structures.)
> 
> Suppose I had sets of structures where I laid them out from left to right
> starting with the smallest, increasing in size until I reached the largest. If
> I had two or more structures the same size, I can place them in any order and
> each different order is counted as a separate structure. The number of
> structures I can create this way is the Somos transform of a[].
> 
> P.S.
> 
> It would be nice if someone (Simon?) could find the definition of Witt vectors.