linear combinations of binomial coefficients

[Max A.]

On 8/2/06, Max A. <maxale at gmail.com> wrote:

> It is interesting to notice that the lowest coefficients seem to form sequence
> A099996(n)=LCM(1,2,...,2n) while the highest coefficients seem to form
> sequence A068550(n)=LCM(1,...,2n)/C(2n,n) with alternating signs. I
> cannot find any other columns/diagonals from this table in the OEIS.

I've guessed a formula for second but last coefficient in the n-th row
of the table. It is
lcm(1,2,...,2*(n-1)) * n / 2 / C(2*(n-2),n-2)

Summarizing, in the n-th row we have coefficients (up to a sign):
[ lcm(1,2,...,2*(n-1)),
... something ...
lcm(1,2,...,2*(n-1)) * n / 2 / C(2*(n-2),n-2),
lcm(1,2,...,2*(n-1)) * / C(2*(n-1),n-1) ]

E.g., for n=10:  [12252240, ..., 4760, 252]

Is an explicit formula for other entries out there?

Max