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%I #4 May 01 2013 21:09:56
%S 2,120,120,2,1436,2,2,5038,5038,2,2,5124,70388,5124,2,2,5238,357640,
%T 357640,5238,2,2,5384,731806,5783216,731806,5384,2,2,5566,1208610,
%U 38702622,38702622,1208610,5566,2,2,5788,1806416,120409892,713559004
%N Triangle read by rows: A(n,k)=A(n - 1, k - 1) + A(n - 1, k) + (n + 1)*(n + 2)*A(n - 2, k - 1).
%C Row sums are (2*(n + 3)!) except for n=1 which is 2: {2, 240, 1440, 10080, 80640, 725760, 7257600, 79833600, 958003200, 12454041600}.
%e {2},
%e {120, 120},
%e {2, 1436, 2},
%e {2, 5038, 5038, 2},
%e {2, 5124, 70388, 5124, 2},
%e {2, 5238, 357640, 357640, 5238, 2},
%e {2, 5384, 731806, 5783216, 731806, 5384, 2},
%e {2, 5566, 1208610, 38702622, 38702622, 1208610, 5566, 2},
%e {2, 5788, 1806416, 120409892, 713559004, 120409892, 1806416, 5788, 2},
%e {2, 6054, 2546916, 281752828, 5942715000, 5942715000, 281752828, 2546916, 6054, 2}
%t Clear[A]; A[2, 1] := A[2, 2] = (5)!;
%t A[3, 2] = 2*(6)! - 4; A[4, 2] = A[4, 3] = (7)! - 2;
%t A[n_, 1] := 2; A[n_, n_] := 2;
%t A[n_, k_] := A[n - 1, k - 1] + A[n - 1, k] + (n + 1)*(n + 2)*A[n - 2, k - 1];
%t a = Table[A[n, k], {n, 10}, {k, n}]; Flatten[a]
%t Table[Apply[Plus, a[[n]]], {n, 1, 10}];
%t Table[Apply[Plus, a[[n]]]/(2*(n + 3)!), {n, 1, 10}]
%K nonn,tabl
%O 1,1
%A _Roger L. Bagula_, Dec 30 2008
%E Edited by _N. J. A. Sloane_, Jun 02 2009
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