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Search: id:A119012
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| A119012 |
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Number of valleys strictly above the x-axis in all hill-free Dyck paths of semilength n (a hill in a Dyck path is a peak at level 1). |
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+0
2
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| 0, 0, 1, 5, 23, 98, 405, 1644, 6604, 26356, 104746, 415155, 1642493, 6490622, 25629581, 101156936, 399151400, 1574818496, 6213255614, 24515233082, 96739530062, 381803092580, 1507141137026, 5950525214360, 23498966702808
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OFFSET
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1,4
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COMMENT
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a(n)=Sum(k*A119011(n,k),k=0..n-2).
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REFERENCES
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E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.
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FORMULA
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G.f.=2[1-3z-(1-z)sqrt(1-4z)]/[(1+2z+sqrt(1-4z))^2*sqrt(1-4z)].
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EXAMPLE
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a(4)=5 because in the 6 (=A000957(5)) hill-free Dyck paths of semilength 4, namely UUUUDDDD, UUUD|UDDD, UUD|UUDDD, UUD|UD|UDD, UUUDD|UDD and UUDDUUDD (U=(1,1), D=(1,-1)) we have altogether 5 valleys strictly above the x-axis (indicated by |).
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MAPLE
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G:=2*(1-3*z-(1-z)*sqrt(1-4*z))/(1+2*z+sqrt(1-4*z))^2/sqrt(1-4*z): Gser:=series(G, z=0, 35): seq(coeff(Gser, z, n), n=1..28);
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CROSSREFS
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Cf. A119011.
Sequence in context: A140529 A055489 A109765 this_sequence A084615 A049674 A077277
Adjacent sequences: A119009 A119010 A119011 this_sequence A119013 A119014 A119015
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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), May 08 2006
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