0,2

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

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Sum_{k=0..n} T(n,k) = A126932(n).

Sum_{k>=0} T(m,k)*T(n,k) = T(m+n,0) = A059738(m+n).

Sum_{k=0..n} T(n,k)*(-k+1) = 3^n. - Philippe Deléham, Mar 26 2007

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;

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 *)

Sequence in context: A141811 A291538 A280787 * A176992 A319375 A107870

Adjacent sequences: A126951 A126952 A126953 * A126955 A126956 A126957

nonn,tabl

Philippe Deléham, Mar 19 2007

approved