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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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified April 19 05:26 EDT 2021. Contains 343105 sequences. (Running on oeis4.)