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A121576
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Riordan array (2-2x-sqrt(1-8x+4x^2),(1-2x-sqrt(1-8x+4x^2))/2).
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5
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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; internal format)
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OFFSET
| 0,2
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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 DELEHAM (kolotoko(AT)wanadoo.fr), Aug 09 2006
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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]
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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 (pbarry(AT)wit.ie), 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)
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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]
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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]
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CROSSREFS
| Sequence in context: A159924 A133367 A179456 * A121575 A049444 A136124
Adjacent sequences: A121573 A121574 A121575 * A121577 A121578 A121579
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Aug 08 2006
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