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A121530
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Number of double rises at an odd 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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2
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0, 1, 4, 14, 47, 148, 454, 1359, 4004, 11644, 33521, 95696, 271300, 764605, 2143964, 5985186, 16643779, 46124692, 127433562, 351106955, 964976460, 2646158176, 7241414949, 19779499584, 53933402472, 146828245753, 399137621524
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| a(n)=Sum(k*A121529(n,k), k>=0). a(n)+A121532(n)=A054444(n-2).
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REFERENCES
| 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
| G.f.=z^2*(1-2z-z^2+4z^3-3z^4)/[(1+z)(1-3z+z^2)^2*(1-z-z^2)].
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EXAMPLE
| a(3)=4 because we have UDUDUD, UDU/UDD, U/UDDUD, U/UDUDD and U/UUDDD, the double rises at an odd level being indicated by a / (U=(1,1), D=(1,-1)).
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MAPLE
| g:=z^2*(1-2*z-z^2+4*z^3-3*z^4)/(1+z)/(1-3*z+z^2)^2/(1-z-z^2): gser:=series(g, z=0, 33): seq(coeff(gser, z, n), n=1..30);
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CROSSREFS
| Cf. A121529, A121532, A054444.
Sequence in context: A049221 A081670 A124805 * A121299 A046718 A192877
Adjacent sequences: A121527 A121528 A121529 * A121531 A121532 A121533
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 05 2006
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