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Search: id:A086617
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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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9
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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
(list; table; graph; listen)
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OFFSET
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0,5
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COMMENT
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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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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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CROSSREFS
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Cf. A086618 (diagonal), A086615 (antidiagonal sums), A003046 (determinants).
Sequence in context: A130671 A114197 A108350 this_sequence A094526 A088699 A101515
Adjacent sequences: A086614 A086615 A086616 this_sequence A086618 A086619 A086620
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KEYWORD
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nonn,tabl
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Jul 24 2003
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EXTENSIONS
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Additional comments from Paul Barry (pbarry(AT)wit.ie), Nov 17 2005
Edited by njas, Oct 16 2006
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