G.f. for a(n) = C(2n,n)*Sum_{k=0..2n} T(n,k)/C(2n,k) ?
Max Alekseyev
maxale at gmail.com
Sun Aug 19 09:31:34 CEST 2007
Somewhat simpler o.g.f. formula:
( 8*sqrt(1-4*t*c)*t*c + 8*sqrt(1-4*t)*t*c - 4*b*sqrt(1-4*t)*t*c -
4*sqrt(1-4*t*c)*b*t - 2*sqrt(1-4*t*c)*c - 2*sqrt(1-4*t) +
b*sqrt(1-4*t) + b*sqrt(1-4*t*c) ) / ( 2 * (1-4*t) * (1-4*t*c) *
(-c-1+4*t*c+b-b^2*t) )
Max
On 8/18/07, Max Alekseyev <maxale at gmail.com> wrote:
> I've got the following o.g.f. for a(n) but I was not able to simplify it much:
>
> SUM a(n) * t^n =
>
> (1+sqrt(1-4*t)) * (-b+sqrt(b^2-4*c)) / sqrt(1-4*t) / ( 8*t*c - 2*b^2*t
> + b^2 + b^2*sqrt(1-4*t) + 2*b*t*sqrt(b^2-4*c) - b*sqrt(b^2-4*c) -
> b*sqrt((b^2-4*c)*(1-4*t)) - b - b*sqrt(1-4*t) - 2*c - 2*sqrt(1-4*t)*c
> + sqrt(b^2-4*c) + sqrt((b^2-4*c)*(1-4*t)) )
>
> +
>
> (1+sqrt(1-4*t*c)) * (-b+sqrt(b^2-4*c)) / sqrt(1-4*t*c) / ( 2*b^2*t -
> 2*b*t*sqrt(b^2-4*c) - 8*t*c - b - b*sqrt(1-4*t*c) + sqrt(b^2-4*c) +
> sqrt((b^2-4*c)*(1-4*t*c)) + 2 + 2*sqrt(1-4*t*c) )
>
> Regards,
> Max
>
> On 8/18/07, Paul D. Hanna <pauldhanna at juno.com> wrote:
> > Seqfans,
> > Can anyone provide a general formula (by inspection) for
> > the g.f. of:
> >
> > (1) a(n) = C(2n,n) * Sum_{k=0..2n} T(n,k) / C(2n,k)
> > where T(n,k) = [x^k] (1 + b*x + c*x^2)^n.
> >
> > See A132310 (copied below) for the special case b=c=1.
> >
> > I suspect that for integer b, c, that the g.f. will be of the form:
> >
> > (2) G.f.: A(x) = 1/sqrt(1 + d*x + e*x^2 + f*x^3).
> >
> > Objective: from integers b, c, in (1) find d, e, and f in (2).
> >
> > I do not know the formula, but (1) generates various sequences
> > already in the OEIS.
> > Thanks,
> > Paul
> > ------------------------------------------------------
> > A132310
> >
> > a(n) = C(2n,n) * Sum_{k=0..2n} trinomial(n,k) / C(2n,k)
> > where trinomial(n,k) = [x^k] (1 + x + x^2)^n.
> >
> > 1,5,21,83,319,1209,4551,17085,64125,240995,907741,3428655,12990121,
> > 49370963,188229489,719805987,2760498351,10615101273,40920439119,
> > 158106581157,612166272291,2374756691313,9228369037659,35918537840577,
> >
> > FORMULA.
> > G.f.: A(x) = 1/sqrt(1 - 10*x + 33*x^2 - 36*x^3).
> >
> > EXAMPLE.
> > a(1) = C(2,1)*(1/1 + 1/2 + 1/1) = 2*(5/2) = 5 ;
> > a(2) = C(4,2)*(1/1 + 2/4 + 3/6 + 2/4 + 1/1) = 6*(7/2) = 21 ;
> > a(3) = C(6,3)*(1/1 + 3/6 + 6/15 + 7/20 + 6/15 + 3/6 + 1/1) = 20*(83/20) =
> > 83.
> >
> > END.
> >
>
* Paul D. Hanna <pauldhanna at juno.com> [Aug 19. 2007 09:13]:
> [...]
>
> BTW, it seems that the OEIS has been down for a while ...
> does anyone know when it will be back up?
>
Note that being unable to reach the server does only
mean something on the way is down:
traceroute to www.research.att.com (192.20.225.32), 30 hops max, 40 byte packets
From:
http://www.exit109.com/~jeremy/news/providers/traceroute.html
------------------------
A trace can end with one of several error indications indicating why
the trace cannot proceed. In this example, the router is indicating
that it has no route to the target host:
The !H is a "host unreachable" error message (it indicates that an
ICMP error message was received). The trace will stop at this
point. Possible ICMP error messages of this nature include:
!H
------------------------
So it may be the server or the router to the net.
The problem above has been there quite a few times now,
IMHO time to act!
> Does this mean that the Seqfan email server is down also?
> Thanks,
> Paul
Let's see if the mail server is affected: let me know if you do not
receive this mail 8-))
>
> [...]
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