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%I #6 Sep 14 2013 17:40:05
%S 2,3,3,2,20,2,2,58,58,2,2,100,516,100,2,2,162,2356,2356,162,2,2,248,
%T 6718,26384,6718,248,2,2,362,16038,165038,165038,16038,362,2,2,508,
%U 34256,664772,2229724,664772,34256,508,2,2,690,67344,2142448,17747916
%N Triangle read by rows: T(n,k) = T(n-1,k-1) +T(n-1,k) +n*(n-1)*T(n-2,k-1) for n>4 and 1<=k<=n.
%F T(1,1)=2. T(2,k)=3. T(3,1) = T(3,3) =T(4,1)= T(4,4)= 2. T(3,2)=20. T(4,2)=T(4,3)=58. Otherwise T(n,k) = T(n-1,k-1) +T(n-1,k) + n*(n-1)*T(n-2,k-1).
%e 2;
%e 3,3;
%e 2,20,2;
%e 2,58,58,2;
%e 2,100,516,100,2;
%e 2,162,2356,2356,162,2;
%e 2,248,6718,26384,6718,248,2;
%e 2,362,16038,165038,165038,16038,362,2;
%e 2,508,34256,664772,2229724,664772,34256,508,2;
%e 2,690,67344,2142448,17747916,17747916,2142448,67344,690,2;
%t Clear[A]; A[2, 1] := A[2, 2] = 3; A[3, 2] = 20;
%t A[4, 2] = 58; A[4, 3] = 58;
%t A[n_, 1] := 2; A[n_, n_] := 2;
%t A[n_, k_] := A[n - 1, k - 1] + A[n - 1, k] + n*(n - 1)*A[n - 2, k - 1];
%t a = Table[A[n, k], {n, 10}, {k, n}];
%t Flatten[a] Table[Apply[Plus, a[[n]]], {n, 1, 10}]; Table[Apply[Plus, a[[n]]]/(n + 1)!, {n, 1, 10}];
%Y Cf. A000142 (row sums).
%K nonn,tabl
%O 1,1
%A _Roger L. Bagula_ and _Gary W. Adamson_, Dec 29 2008
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