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Search: id:A069270
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| A069270 |
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Third level generalization of Catalan triangle (0th level is Pascal's triangle A007318; first level is Catalan triangle A009766; 2nd level is A069269). |
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4
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| 1, 1, 1, 1, 2, 4, 1, 3, 9, 22, 1, 4, 15, 52, 140, 1, 5, 22, 91, 340, 969, 1, 6, 30, 140, 612, 2394, 7084, 1, 7, 39, 200, 969, 4389, 17710, 53820, 1, 8, 49, 272, 1425, 7084, 32890, 135720, 420732, 1, 9, 60, 357, 1995, 10626, 53820, 254475, 1068012, 3362260, 1, 10
(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).
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FORMULA
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T(n, k)=C(n+3k, k)*(n-k+1)/(n+2k+1). For n>=k+3: T(n, k)=T(n-2, k+1)-T(n-3, k+1). T(n, n)=T(n+2, n-1)=C(4n, n)/(3n+1).
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EXAMPLE
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Rows start 1; 1,1; 1,2,4; 1,3,9,22; 1,4,15,52,140; etc.
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CROSSREFS
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Columns include A000012, A000027, A055999. Righthand columns include A002293, A069271, A006632, A002293 (without first 1).
Sequence in context: A158613 A100075 A059836 this_sequence A079901 A121426 A004515
Adjacent sequences: A069267 A069268 A069269 this_sequence A069271 A069272 A069273
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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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