[seqfan] bulk of A046055 (Orders of finite Abelian groups)
Richard Mathar
mathar at strw.leidenuniv.nl
Wed Jul 8 14:39:22 CEST 2009
I am starting to wonder which parts of A046055
http://research.att.com/~njas/sequences/A046055
are correct.
There is a generating function of x(1+2x)/(1-2x) which is
good up to a(17) but then fails--obviously because the pattern of powers of 2
in the actual A046055 then becomes mixed with some stranger numbers like
221184, 442368 etc.
For the same reason, although claimed twice in A046055, the binomial transform of
A010684 does not match, because this is delivering again only the powers of 2
without those 221184 etc.
Similarly, A046055 isn't the row sums of A131127.
Again, the row sums of A134058 are just the powers of 2 (with the exception
of the first row) because A134058 is only twice the Pascal array
Because A093560 [the (3,1) Pascal triangle] contains sums of two rows of the
ordinary Pascal triangle, the row sums of A093560 are again essentially
the powers of two, so that claim in the crossrefs of A046055 does not
look sound either.
The formula in A100320 which says
A046055(n+1)=sum{k=0..floor(n/2), C(n-k,k)(-1)^k*A100320(n-k)}.
is failing for the same reason: the right hand side delivers on the powers of 2.
I think that most of the formulas would have to refer to A000079, A011782,
or A131577, not to A046055, to become correct.
Richard Mathar
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