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A144440 Triangle T(n,k) by rows: T(n, k) = (3*n-3*k+1)*T(n-1, k-1) +(3*k-2)*T(n-1, k) + 3*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.

%I A144440 #9 Mar 04 2022 01:33:21
%S A144440 1,1,1,1,11,1,1,54,54,1,1,229,789,229,1,1,932,7975,7975,932,1,1,3747,
%T A144440 68628,161867,68628,3747,1,1,15010,543144,2534759,2534759,543144,
%U A144440 15010,1,1,60065,4098439,34243778,66389335,34243778,4098439,60065,1
%N A144440 Triangle T(n,k) by rows: T(n, k) = (3*n-3*k+1)*T(n-1, k-1) +(3*k-2)*T(n-1, k) + 3*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
%H A144440 G. C. Greubel, <a href="/A144440/b144440.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A144440 T(n, k) = (3*n-3*k+1)*T(n-1, k-1) +(3*k-2)*T(n-1, k) + 3*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
%F A144440 Sum_{k=1..n} T(n, k) = s(n), where s(n) = (3*n-4)*s(n-1) + 3*s(n-2), s(1) = 1, s(2) = 2.
%F A144440 From _G. C. Greubel_, Mar 03 2022: (Start)
%F A144440 T(n, n-k) = T(n, k).
%F A144440 T(n, 2) = (1/3)*(11*4^(n-2) - (3*n+2)).
%F A144440 T(n, 3) = (1/18)*(9*n^2 + 3*n - 11 - 22*4^(n-3)*(12*n-1) + 709*7^(n-3)). (End)
%e A144440 Triangle begins as:
%e A144440   1;
%e A144440   1,     1;
%e A144440   1,    11,       1;
%e A144440   1,    54,      54,        1;
%e A144440   1,   229,     789,      229,        1;
%e A144440   1,   932,    7975,     7975,      932,        1;
%e A144440   1,  3747,   68628,   161867,    68628,     3747,       1;
%e A144440   1, 15010,  543144,  2534759,  2534759,   543144,   15010,     1;
%e A144440   1, 60065, 4098439, 34243778, 66389335, 34243778, 4098439, 60065, 1;
%t A144440 T[n_, k_, m_, j_]:= T[n,k,m,j]= If[k==1 || k==n, 1, (m*(n-k)+1)*T[n-1,k-1,m,j] + (m*(k-1)+1)*T[n-1,k,m,j] + j*T[n-2,k-1,m,j] ];
%t A144440 Table[T[n,k,3,3], {n,15}, {k,n}]//Flatten (* modified by _G. C. Greubel_, Mar 03 2022 *)
%o A144440 (Sage)
%o A144440 def T(n,k,m,j):
%o A144440     if (k==1 or k==n): return 1
%o A144440     else: return (m*(n-k)+1)*T(n-1,k-1,m,j) + (m*(k-1)+1)*T(n-1,k,m,j) + j*T(n-2,k-1,m,j)
%o A144440 def A144440(n,k): return T(n,k,3,3)
%o A144440 flatten([[A144440(n,k) for k in (1..n)] for n in (1..15)]) # _G. C. Greubel_, Mar 03 2022
%Y A144440 Cf. A144431, A144432, A144435, A144436, A144438, A144439, A144441, A144442, A144443, A144444, A144445.
%K A144440 nonn,tabl
%O A144440 1,5
%A A144440 _Roger L. Bagula_ and _Gary W. Adamson_, Oct 05 2008