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A118973
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Number of hill-free Dyck paths of semilength n+2 and having length of first descent equal to 2 (a hill in a Dyck path is a peak at level 1).
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2
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1, 0, 2, 5, 16, 51, 168, 565, 1934, 6716, 23604, 83806, 300154, 1083137, 3934404, 14374413, 52787766, 194746632, 721435884, 2682522918, 10008240456, 37455101382, 140569122624, 528926230530, 1994980278636, 7541234323096
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Also, for a given j>=2, number of hill-free Dyck paths of semilength n+j and having length of first descent equal to j. a(n)=A000108(n+1)-A000108(n)-[A000957(n+2)-A000957(n+1)]. Columns 2,3,4,... of A118972 (without the initial 0's).
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REFERENCES
| E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241, 241-265, 2001.
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FORMULA
| G.f.=(1-z)CF, where F=[1-sqrt(1-4z)]/[z(3-sqrt(1-4z)] and C=[1-sqrt(1-4z)]/(2z) is the Catalan function.
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EXAMPLE
| a(2)=2 because we have uu(dd)uudd and uuu(dd)udd, 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:=(1-z)*C*F: gser:=series(g, z=0, 33): seq(coeff(gser, z, n), n=0..28);
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CROSSREFS
| Cf. A000108, A000957, A118972, A001558.
Sequence in context: A148384 A148385 A205501 * A148386 A148387 A121651
Adjacent sequences: A118970 A118971 A118972 * A118974 A118975 A118976
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), May 08 2006
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