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Search: id:A126970
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| A126970 |
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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)=T(n-1,1), T(n,k)=T(n-1,k-1)+3*T(n-1,k)+T(n-1,k+1) for k>=1. |
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24
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| 1, 0, 1, 1, 3, 1, 3, 11, 6, 1, 11, 42, 30, 9, 1, 42, 167, 141, 58, 12, 1, 167, 684, 648, 327, 95, 15, 1, 684, 2867, 2955, 1724, 627, 141, 18, 1, 2867, 12240, 13456, 8754, 3746, 1068, 196, 21, 1, 12240, 53043, 61362, 43464, 21060, 7146, 1677, 260, 24, 1
(list; table; graph; listen)
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OFFSET
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0,5
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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)=A126952(n) . Sum_{k, k>=0}T(m,k)*T(n,k)=T(m+n,0)=A117641(m+n).
Sum_{k, 0<=k<=n}T(n,k)*(4*k+1)=5^n . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 22 2007
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EXAMPLE
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Triangle begins:
1;
0, 1;
1, 3, 1;
3, 11, 6, 1;
11, 42, 30, 9, 1;
42, 167, 141, 58, 12, 1;
167, 684, 648, 327, 95, 15, 1;...
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CROSSREFS
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Adjacent sequences: A126967 A126968 A126969 this_sequence A126971 A126972 A126973
Sequence in context: A128162 A067329 A025238 this_sequence A001351 A112811 A040173
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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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