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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

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The n-th row polynomial R(n,x) equals the n-th degree Taylor polynomial of the function (1 - x^2)*(1 + 5*x + x^2)^n expanded about the point x = 0. - Peter Bala, Sep 06 2022

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Riordan array (1/(1 + 5*x + x^2), x/(1 + 5*x + x^2))^(-1); inverse of Riordan array A123967.

T(0)= A053121, T(1)= A064189, T(2)= A039598, T(3)= A091965, T(4)= A052179.

Triangle read by rows: T(n,k) = number of lattice paths from (0,0) to (n,k) that do not go below the line y=0 and consist of steps U=(1,1), D=(1,-1) and five types of steps H=(1,0); example: T(3,1)=77 because we have UDU, UUD, 25 HHU paths, 25 HUH paths and 25 UHH paths . - Philippe Deléham, Sep 25 2007

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 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

Triangle T(5) where T(x) is defined by : T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,k) = T(n-1,k-1) + x*T(n-1,k) + T(n-1,k+1) . Sum_{k, =0<=k<=..n} T(m,k)*T(n,k) = T(m+n,0). Sum_{k, =0<=k<=..n} T(n,k) = A122898(n).

Sum_{k, =0<=k<=..n} T(n,k)*(k+1) = 7^n . - Philippe Deléham, Mar 26 2007

Triangle begins:

1;

5, 1;

26, 10, 1;

140, 77, 15, 1;

777, 540, 153, 20, 1;

4425, 3630, 1325, 254, 25, 1;

25755, 23900, 10509, 2620, 380, 30, 1;

152675, 155764, 79065, 23989, 4550, 531, 35, 1;

919139, 1010560, 575078, 203560, 47270, 7240, 707, 40, 1;

From Philippe Deléham, Nov 07 2011: (Start)

5, 1;

1, 5, 1,;

0, 1, 5, 1;

0, 0, 1, 5, 1;

0, 0, 0, 1, 5, 1;

0, 0, 0, 0, 1, 5, 1;

0, 0, 0, 0, 0, 1, 5, 1;

0, 0, 0, 0, 0, 0, 1, 5, 1;

0, 0, 0, 0, 0, 0, 0, 1, 5, 1; (End)

- From Philippe Deléham, Nov 07 2011

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