Makes sense. (And now I am wondering why I did not see your email until after I replied to the other one. But I guess I am not going to be in a position to solve that particular issue right now.)
Thanks, -- Raul On Tue, Feb 21, 2017 at 2:42 PM, 'Mike Day' via Chat <[email protected]> wrote: > I expect "G.f." means generating function. I should know all about them, > but always lose the thread and forget how they work! > > Running the 2 PARI/GP functions listed is perhaps instructive: > ["(timestamp) gp >" is the prompt in my set-up] > > (19:31) gp > a(n)=sum(k=0, n, binomial(n, k)*(n-k)^k); > (19:32) gp > apply(a,vector(10,i,i)) > %3 = [1, 3, 10, 41, 196, 1057, 6322, 41393, 293608, 2237921] > > (19:32) gp > x='x+O('x^10); Vec( serlaplace( exp(x*exp(x)) ) ) > %4 = [1, 1, 3, 10, 41, 196, 1057, 6322, 41393, 293608] > > (19:33) gp > x > %5 = x + O(x^10) > > (19:33) gp > ?serlaplace \\ built-in help on "serlaplace" > serlaplace(x): replaces the power series sum of a_n*x^n/n! by sum of > a_n*x^n. > For the reverse operation, use serconvol(x,exp(X)). > > Looks as if serlaplace is differentiating the power series... > > Mike > > > On 21/02/2017 18:28, Raul Miller 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, >> > > > --- > This email has been checked for viruses by Avast antivirus software. > https://www.avast.com/antivirus > > > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
