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 A126954 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. 27
 1, 3, 1, 10, 4, 1, 34, 15, 5, 1, 117, 54, 21, 6, 1, 405, 192, 81, 28, 7, 1, 1407, 678, 301, 116, 36, 8, 1, 4899, 2386, 1095, 453, 160, 45, 9, 1, 17083, 8380, 3934, 1708, 658, 214, 55, 10, 1, 59629, 29397, 14022, 6300, 2580, 927, 279, 66, 11, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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 by 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 LINKS G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened FORMULA Sum_{k, 0<=k<=n} T(n,k) = A126932(n). Sum_{k, k>=0} T(m,k)*T(n,k) = T(m+n,0) = A059738(m+n). Sum_{k, 0<=k<=n} T(n,k)*(-k+1) = 3^n. - Philippe Deléham, Mar 26 2007 EXAMPLE Triangle begins: 1; 3, 1; 10, 4, 1; 34, 15, 5, 1; 117, 54, 21, 6, 1; 405, 192, 81, 28, 7, 1; 1407, 678, 301, 116, 36, 8, 1; 4899, 2386, 1095, 453, 160, 45, 9, 1; MATHEMATICA 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 *) CROSSREFS Sequence in context: A141811 A291538 A280787 * A176992 A319375 A107870 Adjacent sequences:  A126951 A126952 A126953 * A126955 A126956 A126957 KEYWORD nonn,tabl AUTHOR Philippe Deléham, Mar 19 2007 STATUS approved

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Last modified October 17 22:41 EDT 2019. Contains 328134 sequences. (Running on oeis4.)