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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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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, 2509378855, 5085153822, 10094528146
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OFFSET
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1,2
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COMMENTS
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See A132890 for the statistic "height" on left factors of Dyck paths.
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LINKS
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FORMULA
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a(n) = Sum_{k=1..n} k * A132890(n,k).
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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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v := ((1-sqrt(1-4*z^2))*1/2)/z: g := proc (k) options operator, arrow: v^k*(1+v)*(1+v^2)/((1+v^(k+1))*(1+v^(k+2))) end proc: T := proc (n, k) options operator, arrow; coeff(series(g(k), z = 0, 70), z, n) end proc: seq(add(k*T(n, k), k = 1 .. n), n = 1 .. 30);
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MATHEMATICA
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b[x_, y_, k_] := b[x, y, k] = If[x == 0, z^k, If[y > 0, b[x - 2, y - 1, k], 0] + b[x - 2, y + 1, Max[y + 1, k]]];
T[n_] := Table[Coefficient[b[2n, 0, 0], z, i], {i, 1, n}];
a[n_] := T[n].Range[n];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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