A121576 Riordan array (2-2*x-sqrt(1-8*x+4*x^2), (1-2*x-sqrt(1-8*x+4*x^2))/2).

1, 2, 1, 6, 5, 1, 24, 24, 8, 1, 114, 123, 51, 11, 1, 600, 672, 312, 87, 14, 1, 3372, 3858, 1914, 618, 132, 17, 1, 19824, 22992, 11904, 4218, 1068, 186, 20, 1, 120426, 140991, 75183, 28383, 8043, 1689, 249, 23, 1, 749976, 884112, 481704, 190347, 58398, 13929, 2508, 321, 26, 1

Offset: 0

Author

Paul Barry, Aug 08 2006

Keywords

• easy
• nonn
• tabl

Comments

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

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

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

Examples

			Triangle begins
     1;
     2,    1;
     6,    5,    1;
    24,   24,    8,   1;
   114,  123,   51,  11,   1;
   600,  672,  312,  87,  14,  1;
  3372, 3858, 1914, 618, 132, 17, 1;
From _Paul Barry_, Apr 27 2009: (Start)
Production matrix is
  2, 1,
  2, 3, 1,
  2, 3, 3, 1,
  2, 3, 3, 3, 1,
  2, 3, 3, 3, 3, 1,
  2, 3, 3, 3, 3, 3, 1,
  2, 3, 3, 3, 3, 3, 3, 1
In general, the production matrix of the inverse of (1/(1-rx),x(1-x)/(1-rx)) is
  -r, 1,
  -r, 1 - r, 1,
  -r, 1 - r, 1 - r, 1,
  -r, 1 - r, 1 - r, 1 - r, 1,
  -r, 1 - r, 1 - r, 1 - r, 1 - r, 1,
  -r, 1 - r, 1 - r, 1 - r, 1 - r, 1 - r, 1,
  -r, 1 - r, 1 - r, 1 - r, 1 - r, 1 - r, 1 - r, 1 (End)
		

Links

• G. C. Greubel, Rows n=0..100 of triangle, flattened

Programs

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

Mathematica

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}]]
(* Emanuele Munarini, May 18 2011 *)

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 */

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

Formula

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