A167630 Riordan array (1/(1-x),xm(x)) where m(x) is the g.f. of Motzkin numbers A001006.

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 8, 8, 4, 1, 1, 17, 20, 13, 5, 1, 1, 38, 50, 38, 19, 6, 1, 1, 89, 126, 107, 63, 26, 7, 1, 1, 216, 322, 296, 196, 96, 34, 8, 1, 1, 539, 834, 814, 588, 326, 138, 43, 9, 1, 1, 1374, 2187, 2236, 1728, 1052, 507, 190, 53, 10, 1

Offset: 0

Author

Philippe Deléham, Nov 07 2009

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  1;
  1,  1;
  1,  2,  1;
  1,  4,  3,  1;
  1,  8,  8,  4,  1;
  1, 17, 20, 13,  5, 1;
  1, 38, 50, 38, 19, 6, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened

Crossrefs

Antidiagonal sums give A082395.

Row sums give A383527.

Diagonals include: A006416, A034856, A086615, A140662.

Cf. A001006, A003462, A064189, A082397, A094531.

Programs

Maple

T:= proc(n, k) option remember; `if`(k=0, 1,
      `if`(k>n, 0, T(n-1, k-1)+T(n-1, k)+T(n-1, k+1)))
    end:
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Apr 20 2018

Mathematica

T[, 0] = T[n, n_] = 1;
T[n_, k_] /; 0, ] = 0;
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 09 2019 *)

Formula

T(n,0)=1, T(0,k)=0 for k>0, T(n,k)=0 if k>n, T(n,k)=T(n-1,k-1)+T(n-1,k)+T(n-1,k+1).

Sum_{k=0..n} k * T(n,k) = A003462(n). - Alois P. Heinz, Apr 20 2018

Sum_{k=0..n} (-1)^(k+1) * T(n,k) = A082397(n-2) for n>=2. - Alois P. Heinz, May 02 2025