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A121576 Riordan array (2-2*x-sqrt(1-8*x+4*x^2), (1-2*x-sqrt(1-8*x+4*x^2))/2). 6
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

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

Sequence in context: A179456 A214152 A121575 * A049444 A136124 A143491

Adjacent sequences:  A121573 A121574 A121575 * A121577 A121578 A121579

KEYWORD

easy,nonn,tabl

AUTHOR

Paul Barry, Aug 08 2006

STATUS

approved

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Last modified March 2 09:56 EST 2021. Contains 341746 sequences. (Running on oeis4.)