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Search: id:A126954
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| A126954 |
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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. |
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+0
25
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| 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; listen)
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OFFSET
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0,2
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COMMENT
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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 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
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FORMULA
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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 DELEHAM (kolotoko(AT)wanadoo.fr), Mar 26 2007
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EXAMPLE
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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;
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CROSSREFS
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Adjacent sequences: A126951 A126952 A126953 this_sequence A126955 A126956 A126957
Sequence in context: A126953 A134284 A134285 this_sequence A107870 A078817 A091042
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KEYWORD
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nonn,tabl
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AUTHOR
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Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 19 2007
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