I reckon that the G.f and E.g.f have very special interpretations. Maybe the Formulas section in https://oeis.org/eishelp2.html might help?
Cheers, Robby On 21 Feb 2017 19:29, "Raul Miller" <[email protected]> wrote: > 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 ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
