Witt vectors A006177 and A006173

[Antti Karttunen]

Wouter Meeussen wrote:
> 
> hi all,
> 
> what follows is a resume of some mails in this forum
> about a year ago (by accident numbered 000108 in seqfan's log)
> 
> the sequence A006973 has a formula for its generation :
>      a(n)=if(n<4,max(n-1,0),
>      (n-1)!*(1+sumdiv(n,k,if(k<n,k*(-a(k)/k!)^(n/k)))))
> 
> its variants A006177 and A006173 do not.
> They carry a cryptic description like "Witt vectors *2!",
> but a simple multipication by 2 is clearly not intended.
> 
> They can be generated from the Catalans as follows:
> 
> build the partitions of n,
> within each partition of n, replace k with z[k] and multiply to get
>     from (3),(2,1),(1,1,1) to (z[3],z[2]z[1], z[1]^3)
> and add these to get somos[3]=z[1]^3+z[1]z[2]+z[3]
> Now get the inverse of this transformation by solving
> a[k]=somos[k] for the z[k].
> 
> cat[n_]:=Binomial[2n,n]/(n+1)
> somos = Table[Plus @@ Apply[Times, Map[z, Partitions[k], {2}], {1}], {k,
> 28}];
> invsomos=Array[z,28]/.Flatten[Last[ Solve[Thread[somos==Array[a,28]],Array[z
> ,28]] ]];
> 
> and then simply apply the inverse somos transform on the catalans:
> 
> invsomos/.a[i_]:>cat[i]
> gives A006177= 1, 1, 3, 8, 25, 72, 245, 772, 2692, 8925
> 
> invsomos/.a[i_]:> cat[1+i]
> gives A006173= 2,1,4,13,44,135,472,1492,5324,17405

How does these transforms, somos and invsomos differ from
the transforms EULER and EULERi given in
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers,
Linear Algebra and Its Applications, vol. 226-228, pp. 57-72, 1995
http://www.research.att.com/~njas/doc/eigen.pdf
but given earlier by
P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102.
where he calls it just "operator S". (on page 91) ?

After I load the transforms implemented in
http://www.research.att.com/~njas/sequences/transforms.txt
to my "student version" of Maple, and do:

A001677 := [1,1,3,8,25,72,245,772,2692,8925,32065,109890,400023,1402723,
            5165327,18484746,68635477,248339122,930138521,3406231198];

EULER(A001677);
[1, 2, 5, 14, 42, 129, 426, 1396, 4811, 16390, 58204, 203845, 736086,

    2626634, 9605866, 34823692, 128620528, 470947139, 1754081722,

    6485357850]

A000108_shifted_left := [seq(Cat(n),n=1..30)];
A000108_shifted_left := [1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796,

    58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790,

    477638700, 1767263190, 6564120420, 24466267020, 91482563640,

    343059613650, 1289904147324, 4861946401452, 18367353072152,

    69533550916004, 263747951750360, 1002242216651368,

    3814986502092304]

EULERi(A000108_shifted_left);

[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, 36054433807398, 136583760011496,

    518401146543811, 1971076356825975]


And the result does not change, even if I apply the transform to the
longer sequence, or make the decimal precision higher.

So, why the differences after the fifth term (129 != 132) in the
first case, and after the seventh term (800 != 772) in the
second case? Either it's not the question of the same transformation
or my Maple implementation is buggy?


Yours,

Antti Karttunen


> 
> Result:
> to expand a series without ability to generate from its original
> description.
> 
> Wouter Meeussen
> ...sine ratio innovo...
> wouter.meeussen at pandora.be