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

0,5

Comments

Diagonal sums : A082395.

Links

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

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

Example

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;
...

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

Crossrefs

Diagonals include: A006416, A034856, A086615, A140662.

Cf. A001006, A003462, A064189, A094531.

Sequence in context: A055587 A137743 A099239 * A322264 A009998 A113993

Adjacent sequences: A167627 A167628 A167629 * A167631 A167632 A167633

Keyword

nonn,tabl

Author

Philippe Deléham, Nov 07 2009

Status

approved