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A121576 Riordan array (2-2x-sqrt(1-8x+4x^2),(1-2x-sqrt(1-8x+4x^2))/2). 5
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

Table of n, a(n) for n=0..50.

FORMULA

T(n,k) = [x^(n-k)](1-2*x-2*x^2)*(1+2*x)^n/(1-x)^(n+1) = 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)/2. [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

Contribution 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]

CROSSREFS

Sequence in context: A133367 A179456 A214152 * A121575 A049444 A136124

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 October 24 03:14 EDT 2014. Contains 248491 sequences.