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A144447 Triangle T(n, k) = T(n-1, k) + T(n, k-1) + T(n-1, k-1) + T(n-2, k-1), with T(n, 1) = T(n, n) = 1, read by rows.

%I A144447 #12 May 23 2023 08:17:35
%S A144447 1,1,1,1,4,1,1,7,13,1,1,10,34,49,1,1,13,64,160,211,1,1,16,103,361,781,
%T A144447 994,1,1,19,151,679,1981,3967,4963,1,1,22,208,1141,4162,10891,20815,
%U A144447 25780,1,1,25,274,1774,7756,24790,60463,112021,137803,1
%N A144447 Triangle T(n, k) = T(n-1, k) + T(n, k-1) + T(n-1, k-1) + T(n-2, k-1), with T(n, 1) = T(n, n) = 1, read by rows.
%H A144447 G. C. Greubel, <a href="/A144447/b144447.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A144447 T(n, k) = T(n-1, k) + T(n, k-1) + T(n-1, k-1) + T(n-2, k-1), with T(n, 1) = T(n, n) = 1.
%F A144447 From _G. C. Greubel_, Mar 09 2022: (Start)
%F A144447 T(n, 2) = (3*n) - 5.
%F A144447 T(n, 3) = (1/2!)*((3*n)^2 - 13*(3*n) + 38).
%F A144447 T(n, 4) = (1/3!)*((3*n)^3 - 24*(3*n)^2 + 195*(3*n) - 606).
%F A144447 T(n, 5) = (1/4!)*((3*n)^4 - 38*(3*n)^3 + 579*(3*n)^2 - 4422*(3*n) + 13704). (End)
%e A144447 Triangle begins as:
%e A144447   1;
%e A144447   1,  1;
%e A144447   1,  4,   1;
%e A144447   1,  7,  13,    1;
%e A144447   1, 10,  34,   49,    1;
%e A144447   1, 13,  64,  160,  211,     1;
%e A144447   1, 16, 103,  361,  781,   994,     1;
%e A144447   1, 19, 151,  679, 1981,  3967,  4963,      1;
%e A144447   1, 22, 208, 1141, 4162, 10891, 20815,  25780,      1;
%e A144447   1, 25, 274, 1774, 7756, 24790, 60463, 112021, 137803, 1;
%t A144447 T[n_, k_]:= T[n,k]= If[k==1 || k==n, 1, T[n-1,k]+T[n,k-1]+T[n-1,k-1]+T[n-2,k-1]];
%t A144447 Table[T[n, k], {n,15}, {k,n}]//Flatten
%o A144447 (Sage)
%o A144447 def T(n,k): return 1 if (k==1 or k==n) else T(n-1, k) + T(n, k-1) + T(n-1, k-1) + T(n-2, k-1) # A144447
%o A144447 flatten([[T(n,k) for k in (1..n)] for n in (1..15)]) # _G. C. Greubel_, Mar 06 2022
%K A144447 nonn,tabl
%O A144447 1,5
%A A144447 _Roger L. Bagula_ and _Gary W. Adamson_, Oct 05 2008
%E A144447 Edited by _G. C. Greubel_, Mar 06 2022