A298646 a(n) is the sum of the degrees of asymmetry of all Dyck paths of semilength n.

0, 0, 4, 18, 88, 360, 1524, 6090, 24784, 98244, 393820, 1556324, 6196656, 24461424, 97079220, 383132250, 1518103840, 5992343940, 23726184372, 93686670220, 370840981680, 1464969055368, 5798679839524, 22917832613988, 90725318348448, 358737952183800

Offset: 1

Author

Emeric Deutsch, Feb 21 2018

Keywords

• nonn

Comments

The degree of asymmetry of a Dyck path is defined in the following manner: we label the steps of a Dyck path of length 2n, from left to right, by 1,2,..., n-1, n, n, n-1, ..., 2,1. The degree of asymmetry is defined to be the number of pairs of identically labeled steps that are not at the same level. Example: the Dyck path uduudd has degree of asymmetry 2. Indeed, the labels are 123321 and the steps labeled 2 are at different levels and those labeled 3 are also at different levels.

All terms are even.

Examples

			a(3) = 4. Indeed, showing the step levels, the 5 = A000108(3) Dyck paths of semilength 3 are 111111, 122221, 123321, 111221, 122111. The first 3 are symmetric (degree of asymmetry 0) and each of the last 2 has degree of asymmetry 2.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1664

Crossrefs

Cf. A000108, A298645.

Programs

Maple

a:= proc(n) option remember; `if`(n<4, [0$3, 4][n+1], (
      2*(n-1)*(34328*n^3+1024539*n^2-2260739*n-3203910)*n*a(n-1)
      +24*(n-1)*(63276*n^4-396683*n^3+460919*n^2+544393*n-1067970)*
      a(n-2)-32*(34328*n^5+784243*n^4-7831140*n^3+24334466*n^2
      -31463717*n+15037140)*a(n-3)-128*(n-3)*(n-4)*(2*n-7)*(38875*n^2
      -225739*n+246552)*a(n-4))/((n+2)*(n+1)*n*(56039*n^2-121145*n-130144)))
    end:
seq(a(n), n=1..30);  # Alois P. Heinz, Feb 21 2018

Mathematica

b[x_, y_, v_] := b[x, y, v] = Expand[If[Min[y, v, x - Max[y, v]] < 0, 0, If[x == 0, 1, Function[l, Sum[Sum[If[y == v + (j - i)/2, 1, z] b[x - 1, y + i, v + j], {i, l}], {j, l}]][{-1, 1}]]]];
g[n_] := g[n] = Sum[b[n, j, j], {j, 0, n}];
T[n_, k_] := Coefficient[g[n], z, k];
a[n_] := Sum[k T[n, k], {k, 0, n - 1}];
Array[a, 30] (* Jean-François Alcover, Dec 21 2020, after Alois P. Heinz in A298645 *)

Formula

a(n) = Sum_{k=0..n-1} k*A298645(n,k).

a(n) = (2*(n-1)*(34328*n^3 + 1024539*n^2 - 2260739*n - 3203910)*n*a(n-1) + 24*(n-1)*(63276*n^4 - 396683*n^3 + 460919*n^2 + 544393*n - 1067970)* a(n-2) - 32*(34328*n^5 + 784243*n^4 - 7831140*n^3 + 24334466*n^2 - 31463717*n + 15037140)*a(n-3) - 128*(n-3)*(n-4)*(2*n-7)*(38875*n^2 - 225739*n + 246552)*a(n-4))/((n+2)*(n+1)*n*(56039*n^2 - 121145*n - 130144)) for n>3, a(n) = 0 for n<3, a(3) = 4. - Alois P. Heinz, Feb 28 2018

a(n) ~ 4^n / sqrt(Pi*n) * (1 - 2/sqrt(Pi*n)). - Vaclav Kotesovec, Mar 06 2018