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A065601
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Number of Dyck paths of length 2n with exactly 1 hill.
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3
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0, 1, 0, 2, 4, 13, 40, 130, 432, 1466, 5056, 17672, 62460, 222853, 801592, 2903626, 10582816, 38781310, 142805056, 528134764, 1960825672, 7305767602, 27307800400, 102371942932, 384806950624, 1450038737668, 5476570993440, 20727983587220, 78606637060012
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OFFSET
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0,4
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COMMENTS
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Convolution of A000957(n) with itself gives a(n-1).
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REFERENCES
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E. Deutsch, Dyck path enumeration, Discrete Math., 204 (1999) 167-202.
E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.
S. Kitaev, J. Remmel and M. Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, Arxiv preprint arXiv:1201.6243, 2012. - From N. J. A. Sloane, May 09 2012
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..500
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FORMULA
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Reference gives g.f.'s.
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MAPLE
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b:= proc(x, y, h, z) option remember;
`if`(x<0 or y<x, 0, `if`(x=0 and y=0, `if`(h, 0, 1),
b(x-1, y, h, is(x=y))+ `if`(h and z, b(x, y-1, false$2),
`if`(z, 0, b(x, y-1, h, false)))))
end:
a:= n-> b(n$2, true$2):
seq (a(n), n=0..30); # Alois P. Heinz, May 10 2012
series(((1-sqrt(1-4*x))/(3-sqrt(1-4*x)))^2/x, x=0, 30); - Mark van Hoeij, Apr 18 2013
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CROSSREFS
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2nd column of A065600. Cf. A000957.
Sequence in context: A033091 A133453 A085422 * A148255 A148256 A163136
Adjacent sequences: A065598 A065599 A065600 * A065602 A065603 A065604
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane, Dec 02 2001
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EXTENSIONS
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More terms from Emeric Deutsch, Dec 03 2001
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STATUS
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approved
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