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A132891
Sum of the heights of all left factors of Dyck paths of length n.
2
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
OFFSET
1,2
COMMENTS
See A132890 for the statistic "height" on left factors of Dyck paths.
LINKS
Toufik Mansour and Gokhan Yilidirim, Longest increasing subsequences in involutions avoiding patterns of length three, Turkish Journal of Mathematics (2019), Section 2.2.
FORMULA
a(n) = Sum_{k=1..n} k * A132890(n,k).
EXAMPLE
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.
MAPLE
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);
MATHEMATICA
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];
Array[a, 33] (* Jean-François Alcover, Nov 10 2020, after Alois P. Heinz in A132890 *)
CROSSREFS
Cf. A132890.
Sequence in context: A356804 A345334 A354294 * A200544 A308448 A055890
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Sep 08 2007
STATUS
approved