(MAGMAMagma) [[(&+[ 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
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reviewed
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proposed
Riordan array (2-2*x-sqrt(1-8*x+4*x^2), (1-2*x-sqrt(1-8*x+4*x^2))/2).
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
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)/2. - Emanuele Munarini, May 18 2011
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,;
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
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approved
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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, 2508, 321, 26, 1
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 _From _Paul Barry_, Apr 27 2009: (Start)
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
-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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editing
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