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A001558
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Number of hill-free Dyck paths of semilength n+3 and having length of first descent equal to 1 (a hill in a Dyck path is a peak at level 1).
(Formerly M2845 N1143)
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5
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1, 3, 10, 33, 111, 379, 1312, 4596, 16266, 58082, 209010, 757259, 2760123, 10114131, 37239072, 137698584, 511140558, 1904038986, 7115422212, 26668376994, 100221202998, 377570383518, 1425706128480, 5394898197448, 20454676622476
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| a(n)=A000957(n+4)-A000957(n+3)-A000957(n+2) (A000957 are the Fine numbers). a(n)=A118972(n+3,1). - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 08 2006
Contribution from Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 05 2008: (Start)
a(n) is also the number of even-length descents to ground level in all Dyck paths of semilength n+2. Example: a(1)=3 because in UDUDUD, UDUU(DD), UU(DD)UD, UUDU(DD) and UUUDDD we have 3 even-length descentts to ground level (shown between parentheses).
a(n)=Sum(k*A111301(n+2,k),k>=0). (End)
Convolution of A000108 with A104629. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 11 2009]
The Kn12 triangle sums of A039599 are given by the terms of this sequence. For the definition of this and other triangle sums see A180662. [From Johannes W. Meijer, Apr 20 2011]
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REFERENCES
| E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241, 241-265, 2001.
T. Fine, Extrapolation when very little is known about the source. Information and Control 16 (1970), 331-359.
S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, < a href="http://arxiv.org/ftp/arxiv/papers/0912/0912.0072.pdf"> Une méthode pour obtenir la fonction génératrice d'une série. FPSAC 1993, Florence. Formal Power Series and Algebraic Combinatorics.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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FORMULA
| G.f. = F*C^3, where F=[1-sqrt(1-4z)]/[z(3-sqrt(1-4z))] and C=[1-sqrt(1-4z)]/(2z) is the Catalan function. - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 08 2006
(n + 3) a(n) = (- 11/2 n + 21/2) a(n - 3) + (9/2 n + 11/2) a(n - 1) + (- 1/2 n + 9/2) a(n - 2) + (- 2 n + 5) a(n - 4). [Simon Plouffe, Feb 09 2012]
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EXAMPLE
| a(1)=3 because we have uu(d)ududd, uuu(d)uddd and uu(d)uuddd, where u=(1,1), d=(1,-1) (the first descents are shown between parentheses).
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MAPLE
| F:=(1-sqrt(1-4*z))/z/(3-sqrt(1-4*z)): C:=(1-sqrt(1-4*z))/2/z: g:=F*C^3: gser:=series(g, z=0, 32): seq(coeff(gser, z, n), n=0..27); - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 08 2006
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CROSSREFS
| Cf. A000957, A118972, A118973.
A111301 [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 05 2008]
Sequence in context: A126931 A071722 A058987 * A111639 A149029 A149030
Adjacent sequences: A001555 A001556 A001557 * A001559 A001560 A001561
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KEYWORD
| nonn,easy,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Edited by Emeric Deutsch (deutsch(AT)duke.poly.edu), May 08 2006
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