Limit in Triangles for Generalized Bell Numbers, Factorials

[Max A.]

On 12/31/06, Paul D. Hanna <pauldhanna at juno.com> wrote:

> (1) Generalized Bell numbers (new triangle A126347):
> B(n,q) = Sum_{k=0..n-1} C(n-1,k)*B(k,q)*q^k for n>0, with B(0,q) = 1.
> and
> (2) Generalized Factorials (new triangle A126470):
> F(n,q) = Sum_{k=0..n-1} C(n-1,k)*F(k,q)*F(n-k-1,q)*q^k for n>0, with
> F(0,q) = 1.
>
> What is interesting is that the reversed rows of the triangles tend to a
> limit.

That's easy to prove. Moreover:

1) Existence of a limit does not depend much on the coefficients in
the recurrent formula for B(n,q) and F(n,q). The only requirement to
the coefficients is that at k=n-1 they must be equal 1 (as C(n-1,k)
does).

2) The m-th leading coefficient in B(n,q) stabilizes at n=m+1. In particular,
A126348(m) and A126471(m) can be defined simply as the coefficient of
q^(m*(m-1)/2) in B(m+1,q) and F(m+1,q) respectively.

[...]

> Row functions B(n,q) in triangle A126347 begin:
> B(0,q) = B(1,q) = 1 ;
> B(1,q) = 1 + q ;
> B(2,q) = 1 + 2*q + q^2 + q^3 ;
> B(3,q) = 1 + 3*q + 3*q^2 + 4*q^3 + 2*q^4 + q^5 + q^6.

I believe it should be
B(0,q) = B(1,q) = 1 ;
B(2,q) = 1 + q ;
B(3,q) = 1 + 2*q + q^2 + q^3 ;
B(4,q) = 1 + 3*q + 3*q^2 + 4*q^3 + 2*q^4 + q^5 + q^6.

[...]

> Row functions F(n,q) of triangle A126470 begin:
> F(0,q) = F(1,q) = 1 ;
> F(1,q) = 1 + q ;
> F(2,q) = 1 + 3*q + q^2 + q^3 ;
> F(3,q) = 1 + 6*q + 7*q^2 + 5*q^3 + 3*q^4 + q^5 + q^6.

Similarly, it should be

F(0,q) = F(1,q) = 1 ;
F(2,q) = 1 + q ;
F(3,q) = 1 + 3*q + q^2 + q^3 ;
F(4,q) = 1 + 6*q + 7*q^2 + 5*q^3 + 3*q^4 + q^5 + q^6.

Max



"David Wilson" <davidwwilson at comcast.net> wrote:
:I think the main reason that programs look strange is that OEIS entries =
:is that the program has to be mangled to fit the OEIS entry format =
:(either stuffed on one line or else split over several lines with fake =
:tabulation characters - neither of which is conducive to downloading and =
:using the program.
:
:How about something like
:
:%p A000008 J. Schmoe, <a =
:href=3Dhttp://www.research.att.com/~njas/sequences/b000008.pl>Perl =
:program</a>
:
:linking to a downloadable text program?

If support for this were added, I'd certainly plan to go through and
replace some of the terse Perl programs I've submitted with something
designed more for legibility.

Hugo



Antti Karttunen <antti.karttunen at gmail.com> wrote:
[...]
:If one considers the same process in factorial base http://www.research.att.com/~njas/sequences/A007623
:then (a similar) sequence can be made infinite. But then only few of these terms
:can be used, because there is an additional condition that the nth digit from
:the right can be at most n. So only 1, 221, 233321, 323321, 332321, 333221, etc. remain.
:and the number of digits must be a triangular number.

The last clause is incorrect: 4444221 is possible.

Hugo