This appears to be a bug in Maxima.
(%i1) load(simplify_sum);
<snip>
(%i3) simplify_sum(sum(binomial(n,k)*binomial(k-1,j)*(-1)**(k-1-j),k,j
+1,n));
(%o3) 0
(%i4) simplify_sum(sum(binomial(5,k)*binomial(k-1,3)*(-1)**(k-1-3),k,
4,5));
(%o4) 1
(%i5) 5*1*1+1*4*(-1);
(%o5) 1
In fact, the answer appears to always be 1 or 0. Is that true?
On Jul 27, 1:17 am, Henryk Trappmann <bo198...@googlemail.com> wrote:
> sage: (n,k,j)=var('n,k,j')
> sage: sum(binomial(n,k)*binomial(k-1,j)*(-1)**(k-1-j),k,j+1,n)
> 0
> sage: (n,j)=(5,3)
> sage: sum(binomial(n,k)*binomial(k-1,j)*(-1)**(k-1-j) for k in range(j
> +1,n+1))
> 1
>
> The symbolic sum being 0 is only trivially valid for n<j+1.
This also discovers another issue:
sage: n = 20
sage: j = 5
sage: var('k')
k
sage: sum(binomial(n,k)*binomial(k-1,j)*(-1)**(k-1-j),k,j+1,n)
TypeError: Either m or x-m must be an integer
Because
sage: binomial(x,3)
1/6*(x - 2)*(x - 1)*x
sage: binomial(3,k)
---------------------------------------------------------------------------
TypeError: Either m or x-m must be an integer
Is this a bug? I would say yes, because
sage: binomial(x,k)
binomial(x, k)
works, but maybe we want to have it be a specific integer if the top
number is given? Any input?
- kcrisman
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