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Search: id:A121532
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| A121532 |
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Number of double rises at an even level in all nondecreasing Dyck paths of semilength 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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3
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| 0, 0, 1, 6, 24, 87, 290, 926, 2861, 8640, 25634, 75015, 217100, 622620, 1772097, 5011394, 14093980, 39448623, 109954398, 305344314, 845165725, 2332485420, 6420202246, 17629525871, 48304680504, 132092031672, 360557665825
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OFFSET
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1,4
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COMMENT
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a(n)=Sum(k*A121531(n,k), k>=0). a(n)+A121530(n)=A054444(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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G.f.=z^3*(1-3z^2+2z^3-z^4)/[(1+z)(1-3z+z^2)^2/(1-z-z^2)].
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EXAMPLE
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a(3)=1 because we have UDUDUD, UDUUDD, UUDDUD, UUDUDD and UU/UDDD, the double rises at an odd level being indicated by a / (U=(1,1), D=(1,-1)).
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MAPLE
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g:=z^3*(1-3*z^2+2*z^3-z^4)/(1+z)/(1-3*z+z^2)^2/(1-z-z^2): gser:=series(g, z=0, 35): seq(coeff(gser, z, n), n=1..32);
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CROSSREFS
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Cf. A121530, A121531, A054444.
Sequence in context: A118043 A166060 A124807 this_sequence A025472 A002919 A006780
Adjacent sequences: A121529 A121530 A121531 this_sequence A121533 A121534 A121535
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 05 2006
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