Extend A131792 ?

[Max Alekseyev]

Paul,

Your program works perfectly in the current CVS version of PARI/GP
(development CHANGES-1.1864).

The following values are obtained in matter of seconds:

1, 1, 2, 6, 21, 76, 280, 1045, 3936, 14925,56892, 217791, 836706,
3224157, 12456225, 48232162, 187131664, 727309265, 2831193004,
11036424667, 43076087806, 168322335246, 658416150496, 2577945422410,
10102468839284, 39621592646545, 155510743708961

Regards,
Max

On 8/30/07, Paul D. Hanna <pauldhanna at juno.com> wrote:
>
>
>
> Seqfans,
>      Would someone be able to extend A131792, which begins:
> 1,1,2,6,21,76,280,1045,3936,14925,56892,217791,836706,3224157,
> 12456225,48232162,
>
> I can get only the next term, 187131664,  before I get "degree overflow"
> using PARI (see program at bottom of this message).
>
> A131792 is the main diagonal of irregular triangle A131791 (see below)
> where A131791(n,k) = the coefficient of x^k of the row g.f. given by:
>
> T(n,k) = [x^k] Product_{j=0..n-1} [ (1 - x^(2^j+1) ) / (1-x) ]
> for n>0, k={0..2^n-1},  with T(0,0)=1.
>
> Of course, diagonal n (below the main diagonal) equals the convolution
> of A000108^n (Catalan sequence to n-th power) and this main diagonal.
>
> Thanks,
>       Paul
> -------------------------------------------------------------------
>
>
> Triangle A131791 begins:
>
> 1;
>
> 1, 1;
>
> 1, 2, 2, 1;
>
> 1, 3, 5, 6, 6, 5, 3, 1;
>
> 1, 4, 9, 15, 21, 26, 29, 30, 30, 29, 26, 21, 15, 9, 4, 1; ...
>
> 1, 5, 14, 29, 50, 76, 105, 135, 165, 194, 220, 241, 256, 265, 269, 270,
>
>  270, 269, 265, 256, 241, 220, 194, 165, 135, 105, 76, 50, 29, 14, 5, 1; ...
>
>
>
> PROGRAM (PARI):
>
> {T(n,k)=if(n==0,1,polcoeff(prod(j=0,n-1,(1-x^(2^j+1))/(1-x)+x*O(x^k)),k))}
>
> /* Print Triangle A131791: */
>
> for(n=0,8,for(k=0,2^n-1,print1(T(n,k),","));print(""))
>
> /* Print Main Diagonal A131792: */
>
> for(n=0,36,print1(T(n,n),","))
>
> END.