Curious binomial-identity /A002720 (small correction of prev. post)

[Gottfried Helms]

Am 25.11.2006 21:10 schrieb Jonathan Post:
> There are several formulae for A002720, including  a(n) = Sum k!C(n,
> k)^2, k=0..n.
> 
> What you write might be true in the asymptotic limit, but not for any
> term, as each term is rational and dividing by e would make each term
> transcendental.  I'm sure that you meant the limit, right?

Yes; as the +... in the numerator should indicate. The sum of columns
have infinitely many terms.

I've seen the generation-formula involving the exp()-function; maybe
this translates in an obvious way (... only with gf's I don't have
*any* experience... sigh...) and the asymptotic formula, which looks
a bit more complicated, but additionally relates it to a pi-expression.

But that we have

   1/3! +  3/2! + 3/1! + 1/0!  = A(3) /3!
                       + 3/1!
                       + 6/2!
                       +10/3!
                       + ...
                        -----
    limit k->oo         A(3) /3! * exp(1)

 or

   1/4! +  4/3! + 6/2! + 4/1! + 1/0!  = A(4)/4!
                              + 4/1!
                              +10/2!
                              +20/3!
                              + ...
                              -----
    limit k->oo               A(4)/4! *exp(1)

has something of a certain beauty... :-)


Gottfried Helms