A283799 Number of dispersed Dyck prefixes of length 2n and height n.

1, 2, 5, 12, 36, 90, 286, 728, 2380, 6120, 20349, 52668, 177100, 460460, 1560780, 4071600, 13884156, 36312408, 124403620, 326023280, 1121099408, 2942885946, 10150595910, 26681566392, 92263734836, 242799302200, 841392966470, 2216352204360, 7694644696200

Offset: 0

Author

Alois P. Heinz, Mar 16 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A282869, A283667, A379822.

Programs

Maple

a:= proc(n) option remember; `if`(n<3, 1+n^2, ((512*(2*n-5))
      *(2519*n-1279)*(n-2)*(2*n-3)*a(n-3) +(192*(2*n-3))
      *(1710*n^3-443*n^2-4990*n+2483)*a(n-2) -(24*(22671*n^4
      -124866*n^3+216436*n^2-129032*n+24526))*a(n-1))
       / ((3*n+2)*(27*n+9)*(855*n-1504)*n))
    end:
seq(a(n), n=0..30);
a := n -> binomial(2*n, n-iquo(n+1, 2)) + binomial(2*n, iquo(n+1,2)-1):
seq(a(n), n = 0..28);  # Peter Luschny, Jan 17 2025

Mathematica

b[x_, y_, m_] := b[x, y, m] = If[x == 0, z^m, If[y > 0, b[x - 1, y - 1, m], 0] + If[y == 0, b[x - 1, y, m], 0] + b[x - 1, y + 1, Max[m, y + 1]]];
a[n_] := Coefficient[b[2n, 0, 0], z, n];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 21 2020, after Alois P. Heinz in A282869 *)

Formula

Recursion: see Maple program.

a(n) = A282869(2*n, n).

From Vaclav Kotesovec, Mar 26 2018: (Start)

Recurrence: 3*n*(3*n + 1)*(3*n + 2)*(3*n^3 - 11*n^2 + 10*n - 3)*a(n) = - 24*(2*n - 1)*(6*n^3 - 1)*a(n-1) + 64*(n-1)*(2*n - 3)*(2*n - 1)*(3*n^3 - 2*n^2 - 3*n - 1)*a(n-2).

a(n) ~ ((3+2*sqrt(3)) - (-1)^n*(3-2*sqrt(3))) * 2^(4*n + 1) / (sqrt(Pi*n) * 3^(3*n/2 + 2)). (End)

From Peter Luschny, Jan 17 2025: (Start)

a(n) = binomial(2*n, n - floor(n/2 + 1/2)) + binomial(2*n, floor(n/2 + 1/2) - 1).

a(n) = A379822(n, (n + 1)/2). (End)