Error in A115866

Dan Dima dimad72 at
Sun Mar 4 00:08:48 CET 2007

Hi all,

I found this more than a year ago when I tried to solve the following puzzle:

However I found a very simple (although infinite sum) formula for the
number of paths from (0,0,...,0) to (a(1),a(2),...,a(k)) using
"nonzero" (2^k-1) steps of the form (x(1),x(2),...,x(k)) where x(i) is
in {0,1} for 1<=i<=k, k-dimensions.

I have looked carefully on the web and I found many articles related
to this issue - Multi-Dimensional Lattice Paths with Diagonal Steps
(or various kind of steps) - but none of them matches my simple
infinite sum:

f(a(1),a(2),...,a(k))  =
Sum( (C(n;a(1)) * C(n;a(2)) * ... C(n;a(k))) / 2^{n+1} , {n,
max(a(1),a(2),...,a(k)), infinity}),
Sum( (C(n;a(1)) * C(n;a(2)) * ... C(n;a(k))) / 2^{n+1} , {n, 0, infinity}),
C(n;a)=n!/a!(n-a)! & we assumed C(n;a)=0 if n<a

Please can someone correct me if I am wrong!

Nick: If you want to compute larger terms for those sequences please
avoid recursivity - a lot of redundant work will be done ;) ... just
use straightforward "for loops" instead...

Neil: what things happen at about 8 or 9 dimensions?

Best regards,

On 3/3/07, N. J. A. Sloane <njas at> wrote:
> David,  strange things happen at about 8 or 9 dimensions,
> so I suggest you go up to dim 10
> of course the arrays and diagonal sequence(s) should
> also be submitted!
> Best
> Neil

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