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Search: id:A121531
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| A121531 |
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Triangle read by rows: T(n,k) is the number of nondecreasing Dyck paths of semilength n and having k double rises at an even level (n>=1, k>=0). 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
3
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| 1, 2, 4, 1, 7, 6, 12, 20, 2, 20, 51, 18, 33, 115, 80, 5, 54, 240, 262, 54, 88, 477, 725, 294, 13, 143, 916, 1803, 1158, 161, 232, 1716, 4170, 3768, 1026, 34, 376, 3155, 9152, 10815, 4684, 475, 609, 5717, 19311, 28418, 17432, 3449, 89, 986, 10240, 39520
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OFFSET
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1,2
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COMMENT
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Row n contains ceil(n/2) terms. Row sums are the odd-subscripted Fibonacci numbers (A001519). T(n,0)=fibonacci(n+2)-1=A000071(n+2). Sum(k*T(n,k), k>=0)=A121532(n).
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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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G=G(t,z)=z(1-2tz^2-tz^3)(1-tz^2)/[(1-z-tz^2)(1-z-z^2-3tz^2-tz^3+t^2*z^4)].
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EXAMPLE
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T(5,2)=2 because we have UU/UU/UDDDDD and UU/UDDU/UDDD, where U=(1,1) and D=(1,-1) (the double rises at an even level are indicated by a /).
Triangle starts:
1;
2;
4,1;
7,6;
12,20,2;
20,51,18;
33,115,80,5;
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MAPLE
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G:=z*(1-2*t*z^2-t*z^3)*(1-t*z^2)/(1-z-t*z^2)/(1-z-z^2-3*t*z^2-t*z^3+t^2*z^4): Gser:=simplify(series(G, z=0, 18)): for n from 1 to 15 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 1 to 15 do seq(coeff(P[n], t, j), j=0..ceil(n/2)-1) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A001519, A121529, A121532.
Sequence in context: A119303 A105552 A112852 this_sequence A127554 A103324 A087060
Adjacent sequences: A121528 A121529 A121530 this_sequence A121532 A121533 A121534
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 05 2006
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