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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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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, 89, 34, 13, 5, 2, 1, 10946
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OFFSET
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1,4
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COMMENTS
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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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LINKS
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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) = t*z*(1-2*t*z)/(1-3*t*z+t^2*z^2)+t*z^2*(1-z)/((1-t*z)* (1-3*z+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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MATHEMATICA
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T[n_, k_] := If[k < n, Fibonacci[2*n - 2*k - 1], Fibonacci[2*n - 3]]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* G. C. Greubel, Oct 22 2017 *)
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PROG
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(PARI) for(n=1, 10, for(k=1, n, print1(if(k<n, fibonacci(2*n-2*k-1), fibonacci(2*n-3)), ", "))) \\ G. C. Greubel, Oct 22 2017
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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