Re: sum of binomials .. bug ?

• To: mathgroup at smc.vnet.net
• Subject: [mg70567] Re: sum of binomials .. bug ?
• From: "Jean-Marc Gulliet" <jeanmarc.gulliet at gmail.com>
• Date: Fri, 20 Oct 2006 05:21:26 -0400 (EDT)
• References: <eh20si\$2ms\$1@smc.vnet.net> <eh4q2f\$8rd\$1@smc.vnet.net> <4537120A.60206@dordos.net>

```On 10/19/06, Peter Pein <petsie at dordos.net> wrote:
> Jean-Marc Gulliet schrieb:
> > yann_che2 at yahoo.fr wrote:
> >> Hi everyone,
> >>
> >> on Mathematica 5.2 (mac os x), experimenting sums of binomials, i tried
> >> the following:
> >>
> >> In[6]:=   f[k_] := Sum[Binomial[21 - k, i], {i, 0, 10 - k}]
> >> In[7]:=   x = 3; f[x]
> >> Out[7]:= 63004
> >> In[8]:=   Clear[x] ; f[x] /. x -> 3
> >> Out[8]:= 262144
> >> In[9]:=   Clear[x] ; f[x]
> >> Out[9]:= 2^(21-x)
> >>
> >>
> >> does anyone know why Out[7] and Out[8] give different results ? do you
> >> think it is a bug ? i searched everywhere in the forums but couldn't
> >> find anything that helped.
> >> do you have a clue ?
> >>
> >> yann
> >>
> > No bug here. You are not evaluating the same function. In the first
> > case, k is replaced by the value 3, then the sum/binomial is evaluated.
> > In the second case, the sum/binomial is evaluated first, then the value
> > 3 is substituted to k. You can get a consistent result using an
> > immediate assignment rather than a delayed one.
> >
> > In[1]:=
> > f[k_] := Sum[Binomial[21 - k, i], {i, 0, 10 - k}]
> >
> > In[2]:=
> > Trace[f[3]]
> >
> > In[3]:=
> > Trace[f[x] /. x -> 3]
> >
> > In[4]:=
> > Clear[f]
> > f[k_] = Sum[Binomial[21 - k, i], {i, 0, 10 - k}]
> >
> > Out[5]=
> > 2^(21 - k)
> >
> > In[6]:=
> > f[3]
> >
> > Out[6]=
> > 262144
> >
> > In[7]:=
> > f[x] /. x -> 3
> >
> > Out[7]=
> > 262144
> >
> > Regards,
> > Jean-Marc
> >
> Hi Jean-Marc,
>
> do not trust any CAS without getting a second opinion. The bugs are not always
> as obvious as in the following example:
>
>
> In[1]:=
> hilDet[n_] := Product[(n^2 - k^2)^(k - n)*k!^2, {k, 1, n - 1}]/n^n;
>
> Testing for some n:
>
> In[2]:=
> And @@ Table[hilDet[k] == Det[Array[1/(#1 + #2 - 1) & , {k, k}]], {k, 10}]
> Out[2]=
> True
>
> split n^2-k^2:
>
> In[3]:=
> bug[n_] := (Product[(n + k)^(k - n)*k!^2, {k, 1, n - 1}]*Product[(n - k)^(k -
> n), {k, 1, n - 1}])/n^n
> In[4]:=
> bug[10] == hilDet[10]
> Out[4]=
> True
>
> well, but:
>
> In[5]:=
> bug[n]
> Out[5]=
> (E^((1/12)*(1 - 6*n + 6*n^2 - 12*Derivative[1, 0][Zeta][-1, 1 - n]))*
>    Product[(k + n)^(k - n)*k!^2, {k, 1, -1 + n}])/(n^n*Glaisher)
> In[6]:=
> % /. n -> 10
> Out[6]=
> (20155392*E^((1/12)*(541 - 12*Derivative[1, 0][Zeta][-1,
> -9])))/(43160654253356787452215625*Glaisher)
> In[7]:=
> hilDet[n]
> Out[7]=
> Product[(-k^2 + n^2)^(k - n)*k!^2, {k, 1, -1 + n}]/n^n
>
> P²
>
Hi Peter,

Interesting example. I agree with you: checking the mathematical
results given by a CAS is a good -- correct -- habit to have. But I
guess I am naturally inclined to trust blindly Mathematica :-) Or
perhaps I just have been for too long on the programming side and not
enough on the mathematical one lately!

Regards,
Jean-Marc

```

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