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A086617
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Symmetric square table of coefficients, read by antidiagonals, where T(n,k) is the coefficient of x^n*y^k in f(x,y) that satisfies f(x,y) = 1/((1-x)(1-y)) + xy*f(x,y)^2.
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8
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1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 7, 4, 1, 1, 5, 13, 13, 5, 1, 1, 6, 21, 33, 21, 6, 1, 1, 7, 31, 69, 69, 31, 7, 1, 1, 8, 43, 126, 183, 126, 43, 8, 1, 1, 9, 57, 209, 411, 411, 209, 57, 9, 1, 1, 10, 73, 323, 815, 1118, 815, 323, 73, 10, 1, 1, 11, 91, 473, 1471, 2633, 2633, 1471, 473, 91, 11, 1
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OFFSET
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0,5
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COMMENTS
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Determinants of upper left n X n matrices results in A003046: {1,1,2,10,140,5880,776160,332972640,476150875200,...}, which is the product of the first n Catalan numbers (A000108).
May also be regarded as a Pascal-Catalan triangle. As a triangle, row sums are A086615, inverse has row sums 0^n.
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LINKS
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FORMULA
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As a triangle, T(n, k)=sum{j=0..n-k, C(n-k, j)C(k, j)C(j)}; T(n, k)=sum{j=0..n, C(n-k, n-j)C(k, j-k)C(j-k)}; T(n, k)=if(k<=n, sum{j=0..n, C(k, j)C(n-k, n-j)C(k-j)}, 0).
As a square array, T(n, k)=sum{j=0..n, C(n, j)C(k, j)C(j)}; As a square array, T(n, k)=sum{j=0..n+k, C(n, n+k-j)C(k, j-k)C(j-k)}; column k has g.f. sum{j=0..k, C(k, j)C(j)(x/(1-x))^j}x^k/(1-x).
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EXAMPLE
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Rows begin:
1, 1, 1, 1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, 6, 7, 8, ...
1, 3, 7, 13, 21, 31, 43, 57, ...
1, 4, 13, 33, 69, 126, 209, 323, ...
1, 5, 21, 69, 183, 411, 815, 1471, ...
1, 6, 31, 126, 411, 1118, 2633, 5538, ...
1, 7, 43, 209, 815, 2633, 7281, 17739, ...
1, 8, 57, 323, 1471, 5538, 17739, 49626, ...
As a triangle:
1;
1, 1;
1, 2, 1;
1, 3, 3, 1;
1, 4, 7, 4, 1;
1, 5, 13, 13, 5, 1;
1, 6, 21, 33, 21, 6, 1;
1, 7, 31, 69, 69, 31, 7, 1;
1, 8, 43, 126, 183, 126, 43, 8, 1;
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MATHEMATICA
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T[n_, k_] := Sum[Binomial[n, j] Binomial[k, j] CatalanNumber[j], {j, 0, n}];
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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