Fwd: [Axiom-developer] sum(binomial(t+i,i),i=0..k)

[Bill Page]

---------- Forwarded message ----------
From: Michael Becker <address@hidden>
Date: Mon, Jan 11, 2010 at 10:18 PM
Subject: Re: [Axiom-developer] sum(binomial(t+i,i),i=0..k)
To: Bill Page <address@hidden>


Am Dienstag, 12. Januar 2010 00:46 schrieb Bill Page:
> What in particular do you think requires explanation? It looks ok to me.
> E.g.
>
> (1) -> ex1:=sum(binomial(t+i,i),i=0..k)
>
>                     t + k      t - 1
>         (t + k + 1)(     ) - t(     )
>                       k         - 1
>    (1)  -----------------------------
>                     t + 1
>                                                      Type: Expression





a)    i do not understand the meaning of binomial(t-1,-1):


 (2) -> kernels(ex1).2

        t - 1
  (2)  (     )
         - 1




b)  i did  expect:   binomial(t+k+1, t+1)



  (3) ->  eval(binomial(t+k+1, t+1),[t=10,k=31])

     (3)  4280561376



c)  axiom does not like  binomial(t-1,-1)


  (4) -> normalize (ex1 - binomial(t+k+1, t+1))

    >> Error detected within library code:
    factorial not defined for negative integers




d)  t*binomial(t-1,-1)  is really = 0  in ex1 :


   (4) -> normalize ( (t+k+1)*binomial(t+k,k)/ (t+1) - binomial(t+k+1, t+1))

    (4)  0









> Integer (2) -> eval(ex1,[t=10,k=31])
>
>    (2)  4280561376
>                                                      Type: Expression
> Integer (3) -> [binomial(10+i,i) for i in 0..31]
>
>    (3)
>    [1, 11, 66, 286, 1001, 3003, 8008, 19448, 43758, 92378, 184756, 352716,
>     646646, 1144066, 1961256, 3268760, 5311735, 8436285, 13123110,
> 20030010, 30045015, 44352165, 64512240, 92561040, 131128140, 183579396,
> 254186856, 348330136, 472733756, 635745396, 847660528, 1121099408]
>                                                            Type: List
> Integer (4) -> reduce(+,%)
>
>    (4)  4280561376
>                                                         Type:
> PositiveInteger
>
> 2010/1/11 Michael Becker <address@hidden>:
> >     can someone explain  the following answer of axiom?
> >
> > (12) -> sum(binomial(t+i,i),i=0..k)
> >
> >                     t + k      t - 1
> >         (t + k + 1)(     ) - t(     )
> >                       k         - 1
> >   (12)  -----------------------------
> >                     t + 1
> >                                                    Type:
> > Expression(Integer)



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