A090182 Triangle T(n,k), 0 <= k <= n, composed of k-Catalan numbers.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 3, 1, 1, 1, 14, 17, 4, 1, 1, 1, 42, 171, 43, 5, 1, 1, 1, 132, 3113, 1252, 89, 6, 1, 1, 1, 429, 106419, 104098, 5885, 161, 7, 1, 1, 1, 1430, 7035649, 25511272, 1518897, 20466, 265, 8, 1, 1, 1, 4862, 915028347, 18649337311, 1558435125, 12833546, 57799, 407, 9, 1, 1

Offset: 0

Author

Philippe Deléham, Jan 20 2004, Oct 16 2008

Keywords

• easy
• nonn
• tabl

Examples

			Triangle begins:
  1;
  1,    1;
  1,    1,       1;
  1,    2,       1,        1;
  1,    5,       3,        1,       1;
  1,   14,      17,        4,       1,     1;
  1,   42,     171,       43,       5,     1,   1;
  1,  132,    3113,     1252,      89,     6,   1, 1;
  1,  429,  106419,   104098,    5885,   161,   7, 1, 1;
  1, 1430, 7035649, 25511272, 1518897, 20466, 265, 8, 1, 1;
This sequence formatted as a square array:
  1, 1, 1,   1,     1,        1,           1,               1, ...
  1, 1, 2,   5,    14,       42,         132,             429, ...
  1, 1, 3,  17,   171,     3113,      106419,         7035649, ...
  1, 1, 4,  43,  1252,   104098,    25511272,     18649337311, ...
  1, 1, 5,  89,  5885,  1518897,  1558435125,   6386478643785, ...
  1, 1, 6, 161, 20466, 12833546, 40130703276, 627122621447281, ...
		

Links

• Alois P. Heinz, Rows n = 0..55, flattened
• Lun Lv, Zhihong Liu, Some Identities Related to Restricted Lattice Paths, 2016 9th International Symposium on Computational Intelligence and Design (ISCID), pp. 338-340.

Crossrefs

The column sequences (without leading zeros) are A000012, A000108 (Catalan), A015083, A015084, A015085, A015086, A015089, A015091, A015092, A015093, A015095, A015096 for k=0..11.

T(2n,n) gives A290777.

Cf. A290759.

Programs

Maple

T:= proc(n, k) option remember; `if`(k=n, 1, add(
      T(j+k, k)*T(n-j-1, k)*k^j, j=0..n-k-1))
    end:
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Aug 10 2017

Mathematica

nmax = 10; col[k_] := col[k] = Module[{A}, A[] = 0; Do[A[x] = Normal[1/(1 - x*A[k*x]) + O[x]^(nmax-k+1)], {nmax-k+1}]; CoefficientList[A[x], x]];
T[n_, k_] := col[k][[n-k+1]];
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 05 2019, using g.f. given for column sequences *)