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Search: id:A118974
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| A118974 |
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Sum of the lengths of the first descents 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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2
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| 0, 0, 2, 4, 11, 31, 94, 298, 977, 3283, 11243, 39087, 137569, 489171, 1754596, 6340756, 23063731, 84372061, 310216081, 1145748061, 4248861631, 15814069951, 59054807821, 221197379221, 830819449003, 3128511421663, 11808294045071
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OFFSET
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0,3
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COMMENT
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a(n)=Sum(k*A118972(n,k),k=1..n).
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REFERENCES
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E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241, 241-265, 2001.
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FORMULA
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G.f.=z^2*CF(1+C-zC)/(1-z), 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
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a(4)=11 because in the hill-free Dyck paths of semilength 4, namely uu(dd)uudd, uu(d)uuddd, uu(d)ududd, uuu(dd)udd, uuu(d)uddd and uuuu(dddd), the sum of the lengths of the first descents (shown between parentheses) is 2+1+1+2+1+4=11.
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MAPLE
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F:=(1-sqrt(1-4*z))/z/(3-sqrt(1-4*z)): C:=(1-sqrt(1-4*z))/2/z: g:=series(z^2*C*F*(1+C-z*C)/(1-z), z=0, 32): seq(coeff(g, z, n), n=0..28);
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CROSSREFS
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Cf. A000957, A118972.
Sequence in context: A148162 A148163 A039300 this_sequence A119020 A073191 A148164
Adjacent sequences: A118971 A118972 A118973 this_sequence A118975 A118976 A118977
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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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