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

0,5

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

Table of n, a(n) for n=0..77.

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