0,2

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 DELEHAM (kolotoko(AT)wanadoo.fr), 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 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 DELEHAM (kolotoko(AT)wanadoo.fr), 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 DELEHAM (kolotoko(AT)wanadoo.fr), 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;

Adjacent sequences: A125903 A125904 A125905 this_sequence A125907 A125908 A125909

Sequence in context: A038243 A075500 A096645 this_sequence A135892 A049460 A062140

nonn,tabl

Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Feb 04 2007