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Search: id:A116914
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| A116914 |
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Number of UUDD's, where U=(1,1) and D=(1,-1), 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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3
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| 1, 1, 5, 16, 58, 211, 781, 2920, 11006, 41746, 159154, 609324, 2341060, 9021559, 34855741, 134972368, 523689718, 2035462990, 7923732118, 30889008112, 120566373676, 471134916286, 1842964183570, 7216096752496, 28279240308268
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OFFSET
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2,3
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COMMENT
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a(n)=Sum(k*A105640(n,k), k=0..floor(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.=z[1+5z-(1-z)sqrt(1-4z)]/[2(2+z)^2*sqrt(1-4z)].
a(n+2)=A126258(2*n,n). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 13 2007
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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 UU(UUDD)DD, UUUDUDDD, UUD(UUDD)D, UUDUDUDD, U(UUDD)UDD, and (UUDD)(UUDD) (U=(1,1), D=(1,-1)) we have altogether 5 UUDD's (shown between parentheses).
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MAPLE
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G:=z*(1+5*z-(1-z)*sqrt(1-4*z))/2/(2+z)^2/sqrt(1-4*z): Gser:=series(G, z=0, 31): seq(coeff(Gser, z^n), n=2..28);
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CROSSREFS
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Cf. A105640.
Sequence in context: A120343 A057553 A006217 this_sequence A047103 A077235 A098347
Adjacent sequences: A116911 A116912 A116913 this_sequence A116915 A116916 A116917
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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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