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Search: id:A112413
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| A112413 |
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Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n and starting with exactly k UD's, where U=(1,1), D=(1,-1) (0<=k<=n). |
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+0
1
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| 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 9, 3, 1, 0, 1, 28, 9, 3, 1, 0, 1, 90, 28, 9, 3, 1, 0, 1, 297, 90, 28, 9, 3, 1, 0, 1, 1001, 297, 90, 28, 9, 3, 1, 0, 1, 3432, 1001, 297, 90, 28, 9, 3, 1, 0, 1, 11934, 3432, 1001, 297, 90, 28, 9, 3, 1, 0, 1, 41990, 11934, 3432, 1001, 297, 90, 28, 9, 3, 1, 0
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OFFSET
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0,7
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COMMENT
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All columns, except for initial terms, yield A000245. Row sums yield the Catalan numbers (A000108).
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FORMULA
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T(n, k)=c(n-k)-c(n-k-1), where c(n)=binomial(2n, n)/(n+1) is the n-th Catalan number. G.f.=(1-z)C/(1-tz), where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.
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EXAMPLE
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T(5,2)=3 because we have UDUDUUDDUD, UDUDUUDUDD and UDUDUUUDDD, where U=(1,1), D=(1,-1).
Triangle begins:
1;
0,1;
1,0,1;
3,1,0,1;
9,3,1,0,1;
28,9,3,1,0,1;
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MAPLE
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T:=proc(n, k) local c: c:=n->binomial(2*n, n)/(n+1): if k<n then c(n-k)-c(n-k-1) elif k=n then 1 else 0 fi end: for n from 0 to 12 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A000108, A000245.
Sequence in context: A151509 A151511 A048993 this_sequence A122960 A091480 A034374
Adjacent sequences: A112410 A112411 A112412 this_sequence A112414 A112415 A112416
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 08 2005
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