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 A086617 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. 8
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS 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. LINKS Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385. FORMULA 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). G.f.: (1-sqrt(1-(4*x^2*y)/((1-x)*(1-x*y))))/(2*x^2*y). - Vladimir Kruchinin, Jan 15 2018 EXAMPLE 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; MATHEMATICA T[n_, k_] := Sum[Binomial[n, j] Binomial[k, j] CatalanNumber[j], {j, 0, n}]; Table[T[n - k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 02 2019 *) CROSSREFS Cf. A086618 (diagonal), A086615 (antidiagonal sums), A003046 (determinants). Sequence in context: A130671 A114197 A108350 * A094526 A088699 A101515 Adjacent sequences:  A086614 A086615 A086616 * A086618 A086619 A086620 KEYWORD nonn,tabl AUTHOR Paul D. Hanna, Jul 24 2003 EXTENSIONS Additional comments from Paul Barry, Nov 17 2005 Edited by N. J. A. Sloane, Oct 16 2006 STATUS approved

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Last modified April 12 17:48 EDT 2021. Contains 342929 sequences. (Running on oeis4.)