A201638 Triangle read by rows, T(n,k) for 0<=k<=n, generalizes the colored Motzkin paths of A107264.

1, 3, 1, 12, 6, 1, 54, 33, 9, 1, 261, 180, 63, 12, 1, 1323, 990, 405, 102, 15, 1, 6939, 5508, 2511, 756, 150, 18, 1, 37341, 30996, 15309, 5229, 1260, 207, 21, 1, 205011, 176256, 92610, 34776, 9630, 1944, 273, 24, 1, 1143801, 1011609, 558414, 225828, 69498, 16281, 2835, 348, 27, 1

Offset

0,2

Links

Table of n, a(n) for n=0..54.

Formula

Recurrence: T(0,0)=1, T(0,k)=0 for k>0 and for n>=1 T(n,k) = T(n-1,k-1)+3*T(n-1,k)+3*T(n-1,k+1).

Example

[0] [1]
[1] [3, 1]
[2] [12, 6, 1]
[3] [54, 33, 9, 1]
[4] [261, 180, 63, 12, 1]
[5] [1323, 990, 405, 102, 15, 1]
[6] [6939, 5508, 2511, 756, 150, 18, 1]
[7] [37341, 30996, 15309, 5229, 1260, 207, 21, 1]

Mathematica

T[0, 0] = 1; T[n_, n_] = 1; T[n_, k_] /; 0 <= k <= n := T[n, k] = T[n - 1, k - 1] + 3*T[n - 1, k] + 3*T[n - 1, k + 1]; T[_, _] = 0;
Table[T[n, k], {n, 0, 9}, {k, 0, n}] (* Jean-François Alcover, Jun 29 2019 *)

Prog

(Sage)
def A201638_triangle(dim):
    T = matrix(ZZ, dim, dim)
    for n in range(dim): T[n, n] = 1
    for n in (1..dim-1):
        for k in (0..n-1):
            T[n, k] = T[n-1, k-1]+3*T[n-1, k]+3*T[n-1, k+1]
    return T
A201638_triangle(8)

Crossrefs

Sequence in context: A122844 A113369 A249253 * A127894 A127898 A078938

Adjacent sequences: A201635 A201636 A201637 * A201639 A201640 A201641

Keyword

nonn,tabl

Author

Peter Luschny, Sep 20 2012

Status

approved