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%I #24 Oct 02 2019 02:54:37
%S 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,
%T 69,69,31,7,1,1,8,43,126,183,126,43,8,1,1,9,57,209,411,411,209,57,9,1,
%U 1,10,73,323,815,1118,815,323,73,10,1,1,11,91,473,1471,2633,2633,1471,473,91,11,1
%N 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.
%C 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).
%C May also be regarded as a Pascal-Catalan triangle. As a triangle, row sums are A086615, inverse has row sums 0^n.
%H Paul Barry, <a href="http://dx.doi.org/10.1016/j.laa.2015.10.032">Riordan arrays, generalized Narayana triangles, and series reversion</a>, Linear Algebra and its Applications, 491 (2016) 343-385.
%F 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).
%F 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).
%F G.f.: (1-sqrt(1-(4*x^2*y)/((1-x)*(1-x*y))))/(2*x^2*y). - _Vladimir Kruchinin_, Jan 15 2018
%e Rows begin:
%e 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 2, 3, 4, 5, 6, 7, 8, ...
%e 1, 3, 7, 13, 21, 31, 43, 57, ...
%e 1, 4, 13, 33, 69, 126, 209, 323, ...
%e 1, 5, 21, 69, 183, 411, 815, 1471, ...
%e 1, 6, 31, 126, 411, 1118, 2633, 5538, ...
%e 1, 7, 43, 209, 815, 2633, 7281, 17739, ...
%e 1, 8, 57, 323, 1471, 5538, 17739, 49626, ...
%e As a triangle:
%e 1;
%e 1, 1;
%e 1, 2, 1;
%e 1, 3, 3, 1;
%e 1, 4, 7, 4, 1;
%e 1, 5, 13, 13, 5, 1;
%e 1, 6, 21, 33, 21, 6, 1;
%e 1, 7, 31, 69, 69, 31, 7, 1;
%e 1, 8, 43, 126, 183, 126, 43, 8, 1;
%t T[n_, k_] := Sum[Binomial[n, j] Binomial[k, j] CatalanNumber[j], {j, 0, n}];
%t Table[T[n - k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Oct 02 2019 *)
%Y Cf. A086618 (diagonal), A086615 (antidiagonal sums), A003046 (determinants).
%K nonn,tabl
%O 0,5
%A _Paul D. Hanna_, Jul 24 2003
%E Additional comments from _Paul Barry_, Nov 17 2005
%E Edited by _N. J. A. Sloane_, Oct 16 2006
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