Some Sum-Of-Sum Congruences
Leroy Quet
qq-quet at mindspring.com
Wed Mar 3 01:40:27 CET 2004
I will just copy/paste my sci.math post.
I only post this message because I assume SOME of you may enjoy it.
I do not know if any of the {A(m,n)} sequences should be in the EIS.
But I will add that the below is a result of:
a(m+1) = sum{k=0 to m} binomial(m,k) (-1)^k a(k),
for all m, m>= 0.
Leroy
qqquet at mindspring.com (Leroy Quet) wrote in message
news:<b4be2fdf.0402281519.269cfb0a at posting.google.com>...
> qqquet at mindspring.com (Leroy Quet) wrote in message
news:<b4be2fdf.0402271732.6a530d54 at posting.google.com>...
> > Let, for each nonnegative integer k,
> >
> > a(6k) = a(6k +1) = 1;
> > a(6k +2) = a(6k +5) = 0;
> > a(6k +3) =a(6k +4) = -1.
> >
> > Let A(0,m) = a(m);
> >
> > and for all positive integers n, and for nonnegative integers m,
> >
> > A(n,m) = sum{k=0 to m} A(n-1,k);
> >
> >
> > Then, for q and r = any nonnegative integers:
> >
> >
> > m!*(A(q,m+1) -binomial(q+m,q-1))
> >
> > is congruent to
> >
> > m!*A(r,m) (-1)^m (mod {m+q+r}).
> >
> >
> > So, more specifically, from the above we get:
> >
> > For ODD m,
> >
> > (m-1)!*A(r,m)
> >
> > is congruent to
> >
> > (r+m)!/(m r!) (mod {m+2r+1}).
> >
> >
> >
> >
> > For EVEN m,
> >
> > (m-1)!*A(r,m)
> >
> > is congruent to
> >
> > (r+m-2)!/(m (r-2)!) (mod {m+2r-2}).
> >
> >
> > (Someone might enjoy confirming the above congruences...)
> >
> > I wonder if any of these congruences have any interesting
> > number-theory implications...
> >
> > thanks,
> > Leroy
> > Quet
>
> I must point out that
>
> A(n,m) =
>
> sum{k>=0} (binomial(m+n-1-6k,n-1) +binomial(m+n-2-6k,n-1)
> -binomial(m+n-4-6k,n-1) -binomial(m+n-5-6k,n-1)),
>
> where
>
> binomial(q,j) = q!/(j!(q-j)!)
> if q >=0, as before,
> but, here,
> binomial(q,j) = 0
> for q < 0.
>
> thanks,
> Leroy Quet
I should probably rewrite the immediately previous sum, for number theory
purposes, as:
For n>= 2,
A(n,m) =
(1/(n-1)!)*
sum{k>=0} sum{j=1 to n-1} S(n-1,j)*(
[m+1-6k]^j +[m-6k]^j -[m-2-6k]^j -[m-3-6k]^j),
where S() is an unsigned Stirling number of the first kind,
and [x] = maximum(0,x).
I should also probably give the first few terms of A(n,m):
A(0,m): 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0,...
A(1,m): 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0,...
A(2,m): 1, 3, 5, 6, 6, 6, 7, 9, 11, 12, 12, 12,...
A(3,m): 1, 4, 9, 15, 21, 27, 34, 43, 54, 66, 78, 90,...
Note that only {A(0,m)} and {A(1,m)} are periodic.
thanks,
Leroy Quet
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