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@math.uqam.ca)
%E A006906 More terms from Vladeta Jovovic (vladeta@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.