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Search: id:A121487
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| A121487 |
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Triangle read by rows: T(n,k) is the number of nondecreasing Dyck paths of semilength n and having abscissa of first return equal to 2k (1<=k<=n). A nondecreasing Dyck path is a Dyck path for which the sequence of the altitudes of the valleys is nondecreasing. |
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+0
1
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| 1, 1, 1, 2, 1, 2, 5, 2, 1, 5, 13, 5, 2, 1, 13, 34, 13, 5, 2, 1, 34, 89, 34, 13, 5, 2, 1, 89, 233, 89, 34, 13, 5, 2, 1, 233, 610, 233, 89, 34, 13, 5, 2, 1, 610, 1597, 610, 233, 89, 34, 13, 5, 2, 1, 1597, 4181, 1597, 610, 233, 89, 34, 13, 5, 2, 1, 4181, 10946, 4181, 1597, 610, 233
(list; table; graph; listen)
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OFFSET
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1,4
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COMMENT
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Row sums are the odd-subscripted Fibonacci numbers (A001519). T(n,1)=T(n,n)=fibonacci(2n-3)=A001519(n-1) for n>=2.
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REFERENCES
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E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.
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FORMULA
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T(n,k)=fibonacci(2n-2k-1) if k<n; T(n,n)=fibonacci(2n-3). G.f.=G(t,z)=tz(1-2tz)/(1-3tz+t^2*z^2)+tz^2*(1-z)/[(1-tz)(1-3z+z^2)].
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EXAMPLE
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T(4,2)=2 because we have UUDDUUDD and UUDDUDUD, where U=(1,1) and D=(1,-1).
Triangle starts:
1;
1,1;
2,1,2;
5,2,1,5;
13,5,2,1,13;
34,13,5,2,1,34;
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MAPLE
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with(combinat): T:=proc(n, k) if k<n then fibonacci(2*n-2*k-1) elif n=k then fibonacci(2*n-3) else 0 fi end: for n from 1 to 13 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A001519.
Sequence in context: A135506 A068822 A090079 this_sequence A057031 A078391 A109631
Adjacent sequences: A121484 A121485 A121486 this_sequence A121488 A121489 A121490
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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), Aug 03 2006
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