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];
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Sep 08 2007
STATUS
approved