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Search: id:A069269
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| A069269 |
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Second level generalization of Catalan triangle (0th level is Pascal's triangle A007318; first level is Catalan triangle A009766; 3rd level is A069270). |
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+0
3
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| 1, 1, 1, 1, 2, 3, 1, 3, 7, 12, 1, 4, 12, 30, 55, 1, 5, 18, 55, 143, 273, 1, 6, 25, 88, 273, 728, 1428, 1, 7, 33, 130, 455, 1428, 3876, 7752, 1, 8, 42, 182, 700, 2448, 7752, 21318, 43263, 1, 9, 52, 245, 1020, 3876, 13566, 43263, 120175, 246675, 1, 10, 63, 320, 1428
(list; table; graph; listen)
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OFFSET
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0,5
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COMMENT
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For the m-th level generalization of Catalan triangle T(n,k)=C(n+mk,k)*(n-k+1)/(n+(m-1)k+1); for n>=k+m: T(n,k)=T(n-m+1,k+1)-T(n-m,k+1); and T(n,n)=T(n+m-1,n-1)=C((m+1)n,n)/(mn+1).
Reflected version of A110616 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 15 2007
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FORMULA
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T(n, k)=C(n+2k, k)*(n-k+1)/(n+k+1). For n>=k+2: T(n, k)=T(n-1, k+1)-T(n-2, k+1). T(n, n)=T(n+1, n-1)=C(3n, n)/(2n+1).
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EXAMPLE
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Rows start 1; 1,1; 1,2,3; 1,3,7,12; 1,4,12,30,55; etc.
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CROSSREFS
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Columns include A000012, A000027, A055998. Righthand columns include (among others) A001764, A006013, A001764 (without first 1), A006629, A006630, A006630, A006631. Cf. triangles A007318, A009766, A069270.
Sequence in context: A062869 A102473 A011117 this_sequence A100324 A121424 A093768
Adjacent sequences: A069266 A069267 A069268 this_sequence A069270 A069271 A069272
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KEYWORD
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nonn,tabl
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AUTHOR
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Henry Bottomley (se16(AT)btinternet.com), Mar 12 2002
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