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.
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
Offset: 0
Examples
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;
Links
- Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.
Programs
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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 *)
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
Extensions
Additional comments from Paul Barry, Nov 17 2005
Edited by N. J. A. Sloane, Oct 16 2006
Comments