linear combinations of binomial coefficients

[Roland Bacher]

Dear Max,

up to a factor, the following formula (in maple notation) seems to
work:

formula:=(n,p)->sum('(-1)^(j+1)*(binomial(2*n-3,n-j)-binomial(2*n-3,n-j-2))*binomial(j*p,p)/j','j'=1..n+1);

A few remarks:

- The above formula yields seemingly
zero for p=1,3,5,...,(2*n-3) (which determines its coefficients k_j
up to a constant factor). Its value at (2n-1) is seemingly given by
(2n-1)^(2n-2)*(-1)^(n+1) (and determines it thus completely).

- Maple extrapolates it for negative integers formula(n,{-1,-2,-3,..})
to the seq 1,2,5,14,.. of Catalan numbers, independently of p. 

- n!*formula(n,0) is seemingly seqnce A71214

-   -1/2*formula(n,2) is seemingly the sequnce 1,1,2,5,14,42,..
(A108) of Catalan numbers

- multiplying the j-th term k_j by j one gets each other row (up to
signs) of the
so-called Catalan-triangle (with entries given by differences of
binomial coefficients). In other terms, replacing the binomial
coeffs binomial(j*p,p)=C(j*p,p) by x^j and derivating, one gets
each other row of the Catalan triangle.

These observations suggest that your observation
has probably some interesting origin.

Roland Bacher



On Thu, Aug 03, 2006 at 01:50:19AM -0700, Max A. wrote:
> 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