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%I #14 Jan 20 2020 21:41:58
%S 1,3,1,10,4,1,34,15,5,1,117,54,21,6,1,405,192,81,28,7,1,1407,678,301,
%T 116,36,8,1,4899,2386,1095,453,160,45,9,1,17083,8380,3934,1708,658,
%U 214,55,10,1,59629,29397,14022,6300,2580,927,279,66,11,1
%N Triangle T(n,k), 0 <= k <= n, read by rows given by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = 3*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-1,k+1) for k >= 1.
%C This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = x*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + y*T(n-1,k) + T(n-1,k+1) for k >= 1. Other triangles arise from choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; (1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906. - _Philippe Deléham_, Sep 25 2007
%H G. C. Greubel, <a href="/A126954/b126954.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F Sum_{k=0..n} T(n,k) = A126932(n).
%F Sum_{k>=0} T(m,k)*T(n,k) = T(m+n,0) = A059738(m+n).
%F Sum_{k=0..n} T(n,k)*(-k+1) = 3^n. - _Philippe Deléham_, Mar 26 2007
%e Triangle begins:
%e 1;
%e 3, 1;
%e 10, 4, 1;
%e 34, 15, 5, 1;
%e 117, 54, 21, 6, 1;
%e 405, 192, 81, 28, 7, 1;
%e 1407, 678, 301, 116, 36, 8, 1;
%e 4899, 2386, 1095, 453, 160, 45, 9, 1;
%t T[0, 0, x_, y_] := 1; T[n_, 0, x_, y_] := x*T[n - 1, 0, x, y] + T[n - 1, 1, x, y]; T[n_, k_, x_, y_] := T[n, k, x, y] = If[k < 0 || k > n, 0, T[n - 1, k - 1, x, y] + y*T[n - 1, k, x, y] + T[n - 1, k + 1, x, y]]; Table[T[n, k, 3, 1], {n, 0, 10}, {k, 0, n}] // Flatten (* _G. C. Greubel_, May 22 2017 *)
%K nonn,tabl
%O 0,2
%A _Philippe Deléham_, Mar 19 2007
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