0,4

Similar to A064189 (x = 1) and to A039599 (x = 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 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

Row sums yield A126568 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 10 2007

T(n, 0) = A033321(n) and for k>=1: T(n, k) = Sum_{j, j>=1} T(n-j, k-1)*A002212(j).

Sum_{k, 0<=k<=n} T(m, k)*T(n, k) = T(m+n, 0) = A033321(m+n).

The triangle may also be generated from M^n * [1,0,0,0...], where M = an infinite tridiagonal matrix with 1's in the super and subdiagonals and (1,3,3,3...) in the main digonal. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 17 2006

Sum_{k, 0<=k<=n}T(n,k)*(3*k+1)=5^n . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Feb 26 2007

Sum{k, 0<=k<=n}T(n,k)*(3*k+1)=5^n . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 25 2007

Sum_{k, 0<=k<=n}T(n,k)=A126568(n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 10 2007

Triangle begins:

1;

1, 1;

2, 4, 1;

6, 15, 7, 1;

21, 58, 37, 10, 1;

79, 232, 179, 68, 13, 1;

311, 954, 837, 396, 108, 16, 1;

1265, 4010, 3861, 2133, 736, 157, 19, 1;

5275, 17156, 17726, 10996, 4498, 1226, 215, 22, 1;

22431, 74469, 81330, 55212, 25716, 8391, 1893, 282, 25, 1;

...

Cf. A002212, A033321, A039599, A064189.

The inverse of A126126.

Sequence in context: A105357 A167546 A011369 this_sequence A021009 A137478 A089087

Adjacent sequences: A110874 A110875 A110876 this_sequence A110878 A110879 A110880

nonn,easy,tabl

Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 19 2005