I was looking at https://oeis.org/A000248 (because of its relevance to
idempotence), and I ran into some problems understanding the formula.
One of them makes sense to me:
a(n) = Sum_{k=0..n} C(n,k)*(n-k)^k. [Paul D. Hanna, Jun 26 2009]
corresponds to:
k=:i.@>:
+/@((!~k)*(-k)^k)"0 i.10
1 1 3 10 41 196 1057 6322 41393 293608
But the two preceding that give me problems.
For example, I look at E.g.f.: exp(x*exp(x)) and that seems to me to represent:
^(* ^) i.10
1 15.1543 2.6185e6 1.47609e26 7.02589e94 _ _ _ _ _
I do not see how that can ever be relevant. But, ok, maybe I need an
integer base for the exponent. The only integer which gets me "closer"
to the desired sequence would be 2, so:
2&^(* 2&^) i.10
1 4 256 1.67772e7 1.84467e19 1.4615e48 3.9402e115 5.28295e269 _ _
... that still does not make sense to me. I don't even know why that
formula is there. Maybe I need to be using some different value for x?
But I doubt it, because the growth rate looks wrong for both of those
sequences.
And, the next one:
G.f.: Sum_{k>=0} x^k/(1-k*x)^(k+1). - Vladeta Jovovic, Oct 25 2003
This one also seems like garbage - there's two variables here, and
there's no constraint that tells me about whether it's x or k that is
supposed to correspond to the index position in the sequence, and
likewise there's nothing that tells me what the other value should be.
Or, ok, maybe that's supposed to be an infinite sequence in k which
converges (and x is the index position)? Let's try that:
k=:i.10
3 :'+/y^k%(1-k*y)^k+1'"0 ] i.10
1 10 10.9531 10.2994 10.1588 10.1015 10.0717 10.0538 10.0421 10.034
k=:i.100
3 :'+/y^k%(1-k*y)^k+1'"0 ] i.10
1 100 100.953 100.299 100.159 100.102 100.072 100.054 100.042 100.034
k=:i.1000
3 :'+/y^k%(1-k*y)^k+1'"0 ] i.10
1 1000 1000.95 1000.3 1000.16 1000.1 1000.07 1000.05 1000.04 1000.03
Unless I have made a major mistake, it looks like that is not a useful
interpretation of that formula.
Then again, maybe I am overlooking some quirk of notation? I only was
able to make sense of the Paul D. Hanna formula because I recognized
the C(n,k) as what we would express in J as k!n
So... since I know some other people here have stronger backgrounds in
this kind of thing than I - am I overlooking something important here?
I'd really prefer to be able to understand what I read.
Thanks,
--
Raul
----------------------------------------------------------------------
For information about J forums see http://www.jsoftware.com/forums.htm
