%I #20 Sep 08 2022 08:45:27

%S 1,2,1,6,5,1,24,24,8,1,114,123,51,11,1,600,672,312,87,14,1,3372,3858,

%T 1914,618,132,17,1,19824,22992,11904,4218,1068,186,20,1,120426,140991,

%U 75183,28383,8043,1689,249,23,1,749976,884112,481704,190347,58398,13929,2508,321,26,1

%N Riordan array (2-2*x-sqrt(1-8*x+4*x^2), (1-2*x-sqrt(1-8*x+4*x^2))/2).

%C Inverse of Riordan array (1/(1+2*x), x*(1-x)/(1+2*x)).

%C Row sums are A047891; first column is A054872. Signed version given by A121575.

%C Triangle T(n,k), 0 <= k <= n, read by rows, given by [2, 1, 3, 1, 3, 1, 3, 1, 3, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Aug 09 2006

%H G. C. Greubel, <a href="/A121576/b121576.txt">Rows n=0..100 of triangle, flattened</a>

%F T(n,k) = [x^(n-k)](1-2*x-2*x^2)*(1+2*x)^n/(1-x)^(n+1) = (1/2)*Sum_{i=0..n-k} binomial(n,i) * binomial(2*n-k-i,n) * (4 - 9*i + 3*i^2 - 6*(i-1)*n + 2*n^2)/((n-i+2)*(n-i+1))*2^i. - _Emanuele Munarini_, May 18 2011

%e Triangle begins

%e 1;

%e 2, 1;

%e 6, 5, 1;

%e 24, 24, 8, 1;

%e 114, 123, 51, 11, 1;

%e 600, 672, 312, 87, 14, 1;

%e 3372, 3858, 1914, 618, 132, 17, 1;

%e From _Paul Barry_, Apr 27 2009: (Start)

%e Production matrix is

%e 2, 1,

%e 2, 3, 1,

%e 2, 3, 3, 1,

%e 2, 3, 3, 3, 1,

%e 2, 3, 3, 3, 3, 1,

%e 2, 3, 3, 3, 3, 3, 1,

%e 2, 3, 3, 3, 3, 3, 3, 1

%e In general, the production matrix of the inverse of (1/(1-rx),x(1-x)/(1-rx)) is

%e -r, 1,

%e -r, 1 - r, 1,

%e -r, 1 - r, 1 - r, 1,

%e -r, 1 - r, 1 - r, 1 - r, 1,

%e -r, 1 - r, 1 - r, 1 - r, 1 - r, 1,

%e -r, 1 - r, 1 - r, 1 - r, 1 - r, 1 - r, 1,

%e -r, 1 - r, 1 - r, 1 - r, 1 - r, 1 - r, 1 - r, 1 (End)

%t Flatten[Table[Sum[Binomial[n,i]Binomial[2n-k-i,n](4-9i+3i^2-6(i-1)n+2n^2)/((n-i+2)(n-i+1))2^i,{i,0,n-k}]/2,{n,0,8},{k,0,n}]]

%t (* _Emanuele Munarini_, May 18 2011 *)

%o (Maxima) create_list(sum(binomial(n,i)*binomial(2*n-k-i,n)*(4-9*i+3*i^2-6*(i-1)*n+2*n^2)/((n-i+2)*(n-i+1))*2^i,i,0,n-k)/2,n,0,8,k,0,n); /* Emanuele Munarini, May 18 2011 */

%o (PARI) for(n=0,10, for(k=0,n, print1(sum(j=0, n-k, 2^j*binomial(n,j) *binomial(2*n-k-j, n)*(4-9*j+3*j^2-6*(j-1)*n + 2*n^2)/((n-j+2)*(n-j+1)))/2, ", "))) \\ _G. C. Greubel_, Nov 02 2018

%o (Magma) [[(&+[ 2^j*Binomial(n,j)*Binomial(2*n-k-j, n)*(4-9*j+3*j^2-6*(j-1)*n + 2*n^2)/((n-j+2)*(n-j+1))/2: j in [0..(n-k)]]): k in [0..n]]: n in [0..10]]; // _G. C. Greubel_, Nov 02 2018

%K easy,nonn,tabl

%O 0,2

%A _Paul Barry_, Aug 08 2006