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A201638
Triangle read by rows, T(n,k) for 0<=k<=n, generalizes the colored Motzkin paths of A107264.
0
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
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
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Sep 20 2012
STATUS
approved