OFFSET
0,2
COMMENTS
Inverse of Riordan array (1/(1+2*x), x*(1-x)/(1+2*x)).
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
LINKS
G. C. Greubel, Rows n=0..100 of triangle, flattened
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
EXAMPLE
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)
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 *)
PROG
(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
(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
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, Aug 08 2006
STATUS
approved