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A126953
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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+1) for k>=1.
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24
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1, 3, 1, 10, 3, 1, 33, 11, 3, 1, 110, 36, 12, 3, 1, 366, 122, 39, 13, 3, 1, 1220, 405, 135, 42, 14, 3, 1, 4065, 1355, 447, 149, 45, 15, 3, 1, 13550, 4512, 1504, 492, 164, 48, 16, 3, 1, 45162, 15054, 5004, 1668, 540, 180, 51, 17, 3, 1
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OFFSET
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0,2
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COMMENTS
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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, Sep 25 2007
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LINKS
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Table of n, a(n) for n=0..54.
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FORMULA
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Sum_{k, 0<=k<=n}T(n,k)=A127359(n) . Sum_{k, k>=0}T(m,k)*T(n,k)=T(m+n,0)=A126931(m+n).
Sum_{k, 0<=k<=n}T(n,k)*(-2*k+1)=2^n . - Philippe DELEHAM, Mar 25 2007
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EXAMPLE
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Triangle begins:
1;
3, 1;
10, 3, 1;
33, 11, 3, 1;
110, 36, 12, 3, 1;
366, 122, 39, 13, 3, 1;
1220, 405, 135, 42, 14, 3, 1;
4065, 1355, 447, 149, 45, 15, 3, 1;
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CROSSREFS
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Sequence in context: A016478 A102430 A135573 * A134284 A134285 A141811
Adjacent sequences: A126950 A126951 A126952 * A126954 A126955 A126956
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KEYWORD
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nonn,tabl
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AUTHOR
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Philippe DELEHAM, Mar 19 2007
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STATUS
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approved
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