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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

See A132890 for the statistic "height" on left factors of Dyck paths.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..700

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-Fran├žois Alcover, Nov 10 2020, after Alois P. Heinz in A132890 *)

CROSSREFS

Cf. A132890.

Sequence in context: A006951 A224840 A345334 * A200544 A308448 A055890

Adjacent sequences:  A132888 A132889 A132890 * A132892 A132893 A132894

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Sep 08 2007

STATUS

approved

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Last modified July 30 19:33 EDT 2021. Contains 346359 sequences. (Running on oeis4.)