Witt vectors A006177 and A006173
Wouter Meeussen
wouter.meeussen at pandora.be
Tue Aug 13 22:13:35 CEST 2002
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
Result:
to expand a series without ability to generate from its original
description.
Wouter Meeussen
...sine ratio innovo...
wouter.meeussen at pandora.be
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