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Search: id:A132891
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| A132891 |
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Sum of the heights of all left factors of Dyck paths of length n. |
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+0
2
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| 1, 3, 6, 14, 28, 61, 121, 257, 508, 1065, 2103, 4372, 8634, 17842, 35254, 72524, 143396, 293968, 581630, 1189102, 2354168, 4802331, 9512984, 19370764, 38391332, 78056544, 154773135, 314281350, 623427154, 1264546021
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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See A132890 for the statistic "height" on left factors of Dyck paths.
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FORMULA
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a(n)=Sum(k*A132890(n,k),k=1..n).
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EXAMPLE
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a(4)=14 because the six left factors of Dyck paths of length 4 are UDUD, UDUU, UUDD, UUDU, UUUD and UUUU, having heights 1, 2, 2, 2, 3 and 4, respectively.
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MAPLE
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H[0]:=1: for n to 40 do H[n]:=simplify(1/(1-z^2*H[n-1])) end do: G[0]:=1: for n to 40 do G[n]:=simplify(H[n]+z*H[n]*G[n-1]) end do: g[0]:=G[0]: for n to 40 do g[n]:= simplify(G[n]-G[n-1]) end do: for n from 0 to 40 do gser[n]:=series(g[n], z=0, 50) end do: for n to 40 do for k to n do T:=proc(n, k) options operator, arrow: coeff(gser[k], z, n) end proc end do end do: seq(add(k*T(n, k), k=1..n), n=1..30);
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CROSSREFS
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Cf. A132890.
Sequence in context: A030012 A001970 A006951 this_sequence A055890 A038359 A038360
Adjacent sequences: A132888 A132889 A132890 this_sequence A132892 A132893 A132894
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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), Sep 08 2007
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