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

%I A201638 #14 Mar 17 2020 14:21:10
%S A201638 1,3,1,12,6,1,54,33,9,1,261,180,63,12,1,1323,990,405,102,15,1,6939,
%T A201638 5508,2511,756,150,18,1,37341,30996,15309,5229,1260,207,21,1,205011,
%U A201638 176256,92610,34776,9630,1944,273,24,1,1143801,1011609,558414,225828,69498,16281,2835,348,27,1
%N A201638 Triangle read by rows, T(n,k) for 0<=k<=n, generalizes the colored Motzkin paths of A107264.
%F A201638 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).
%e A201638 [0] [1]
%e A201638 [1] [3, 1]
%e A201638 [2] [12, 6, 1]
%e A201638 [3] [54, 33, 9, 1]
%e A201638 [4] [261, 180, 63, 12, 1]
%e A201638 [5] [1323, 990, 405, 102, 15, 1]
%e A201638 [6] [6939, 5508, 2511, 756, 150, 18, 1]
%e A201638 [7] [37341, 30996, 15309, 5229, 1260, 207, 21, 1]
%t A201638 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;
%t A201638 Table[T[n, k], {n, 0, 9}, {k, 0, n}] (* _Jean-François Alcover_, Jun 29 2019 *)
%o A201638 (Sage)
%o A201638 def A201638_triangle(dim):
%o A201638     T = matrix(ZZ,dim,dim)
%o A201638     for n in range(dim): T[n,n] = 1
%o A201638     for n in (1..dim-1):
%o A201638         for k in (0..n-1):
%o A201638             T[n,k] = T[n-1,k-1]+3*T[n-1,k]+3*T[n-1,k+1]
%o A201638     return T
%o A201638 A201638_triangle(8)
%K A201638 nonn,tabl
%O A201638 0,2
%A A201638 _Peter Luschny_, Sep 20 2012