%I #4 Aug 09 2015 01:11:04

%S 2,9,9,2,212,2,2,1618,1618,2,2,2100,54116,2100,2,2,2786,609572,609572,

%T 2786,2,2,3712,1582558,26220736,1582558,3712,2,2,4914,3257870,

%U 393546494,393546494,3257870,4914,2,2,6428,6069056,1593218212,20609969404

%N A triangular sequence with row sums (3^(n - 1)*(n + 1)!) starting at n=1 which was calculated by steps.

%C Row sums are 3^(n - 1)*(n + 1)!.

%C A fractal plot is:

%C a0 = Table[Table[If[m <= n, If[Mod[A[n, m], 3] == 0, 0, 1], 0], {m, 1, 12}], {n, 1, 12}];

%C ListDensityPlot[a0, Mesh -> False, Axes -> False]

%F A(n,k)=A(n - 1, k - 1) + A(n - 1, k) + b[n]*n*(n + 1)*A(n - 2, k - 1);

%F b[n] is an array function of n.

%e {2},

%e {9, 9},

%e {2, 212, 2},

%e {2, 1618, 1618, 2},

%e {2, 2100, 54116, 2100, 2},

%e {2, 2786, 609572, 609572, 2786, 2},

%e {2, 3712, 1582558, 26220736, 1582558, 3712, 2},

%e {2, 4914, 3257870, 393546494, 393546494, 3257870, 4914, 2},

%e {2, 6428, 6069056, 1593218212, 20609969404, 1593218212, 6069056, 6428, 2},

%e {2, 8290, 10645504, 4629106368, 388201427036, 388201427036, 4629106368, 10645504, 8290, 2},

%e {2, 10536, 17866010, 11449232704, 2180421367268, 23900788525360, 2180421367268, 11449232704, 17866010, 10536, 2}

%t Clear[a]; a = {{2}, {9, 9}, {2, 212, 2}, {2, 1618, 1618, 2},

%t {2, 2100, 54116, 2100, 2}, {2, 2786, 609572, 609572, 2786, 2},

%t {2, 3712, 1582558, 26220736, 1582558, 3712, 2}, {2,4914, 3257870, 393546494, 393546494, 3257870, 4914, 2},

%t {2, 6428, 6069056, 1593218212, 20609969404, 1593218212, 6069056, 6428, 2},

%t {2, 8290, 10645504, 4629106368, 388201427036, 388201427036, 4629106368, 10645504, 8290, 2},

%t {2, 10536, 17866010, 11449232704, 2180421367268, 23900788525360,2180421367268, 11449232704, 17866010, 10536, 2}};

%t Flatten[a] Table[Apply[Plus, a[[n]]], {n, 1, Length[a]}];

%t Table[Apply[Plus, a[[n]]]/(3^(n - 1)*(n + 1)!), {n, 1, Length[a]}];

%t Clear[A, b]; Table[b[n] = (39 n + 9 n^2)/(n + 1), {n, 1, 4}];

%t b[5] = 8; b[6] = 57/7; b[7] = 33/4; b[8] = 25/3; b[9] = 42/5;

%t b[10] = 93/11; b[11] = 17/2; b[12] = 111/13;

%t A[2, 1] := A[2, 2] = 9; A[3, 2] = 212;

%t A[4, 2] = 1618; A[4, 3] = 1618;

%t A[n_, 1] := 2; A[n_, n_] := 2;

%t A[n_, k_] := A[n - 1, k - 1] + A[n - 1, k] + b[n]*n*(n + 1)*A[n - 2, k - 1];

%t a = Table[A[n, k], {n, 12}, {k, n}];

%t Flatten[a]

%t Table[Apply[Plus, a[[n]]], {n, 1, 12}];

%t Table[Apply[Plus, a[[n]]]/(3^(n - 1)*(n + 1)!), {n, 1, 12}];

%K nonn,uned,tabl

%O 1,1

%A _Roger L. Bagula_ and _Gary W. Adamson_, Dec 29 2008, Jan 01 2009