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Search: id:A101975
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| A101975 |
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Triangle read by rows: number of Dyck paths of semilength n with k peaks after the first return (0<= k <n). |
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+0
2
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| 1, 1, 1, 2, 2, 1, 5, 4, 4, 1, 14, 9, 11, 7, 1, 42, 23, 27, 28, 11, 1, 132, 65, 66, 87, 62, 16, 1, 429, 197, 170, 239, 250, 122, 22, 1, 1430, 626, 471, 627, 829, 630, 219, 29, 1, 4862, 2056, 1398, 1656, 2448, 2553, 1419, 366, 37, 1, 16796, 6918, 4381, 4554, 6803, 8813
(list; table; graph; listen)
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OFFSET
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1,4
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REFERENCES
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E. Deutsch, Dyck path enumeration, Discrete Math., 204, 1999, 167-202.
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FORMULA
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T(n, 0)=c(n-1), T(n, 1) = sum(c(i), i=0..n-2), T(n, k)= sum(c(j)*binomial(n-1-j, k-1)*binomial(n-1-j, k)/(n-1-j), j=0..n-2) for k>=2, where c(i)=binomial(2i, i)/(i+1) (i=0, 1, ...) are the Catalan numbers (A000108).
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EXAMPLE
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T(4,2)=4 because we have UD|(UD)U(UD)D, UD|U(UD)D(UD), UD|U(UD)(UD)D, and
UUDD|(UD)(UD), where U=(1,1), D=(1,-1) (the two peaks after the first return | are shown between parentheses).
Triangle begins:
1;
1,1;
2,2,1;
5,4,4,1;
14,9,11,7,1;
42,23,27,28,11,1;
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MAPLE
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c:=n->binomial(2*n, n)/(n+1):T:=proc(n, k) if k=0 then c(n-1) elif k=1 then sum(c(i), i=0..n-2) else sum(c(j)*binomial(n-1-j, k-1)*binomial(n-1-j, k)/(n-1-j), j=0..n-2) fi end: for n from 1 to 11 do seq(T(n, k), k=0..n-1) od; # yields the sequence in triangular form
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CROSSREFS
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Cf. A000108, A014137, A101974.
Sequence in context: A105292 A125177 A125178 this_sequence A136388 A099605 A079218
Adjacent sequences: A101972 A101973 A101974 this_sequence A101976 A101977 A101978
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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 22 2004
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