A283667 Number of Motzkin prefixes of length 2n and height n.

1, 3, 13, 64, 334, 1802, 9933, 55575, 314362, 1793126, 10295625, 59430043, 344559826, 2005026610, 11703965955, 68503652100, 401892122682, 2362629703214, 13914547415998, 82081163986020, 484893156220356, 2868234297838092, 16986185485228431, 100703275233924096

Offset: 0

Author

Alois P. Heinz, Mar 13 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1279

Crossrefs

Cf. A283595, A283799.

Programs

Maple

a:= proc(n) option remember; `if`(n<2, 1+2*n, ((4*n-2)*(455*n^6-
       1155*n^5-2776*n^4+1047*n^3+1493*n^2-72*n-72)*a(n-1)+36*
      (n-1)*(2*n-1)*(2*n-3)*(13*n^4-7*n^3-64*n^2-44*n-6)*a(n-2))/
      ((9*n+6)*(3*n+1)*(n+1)*(13*n^4-59*n^3+35*n^2+11*n-6)))
    end:
seq(a(n), n=0..30);

Mathematica

b[x_, y_, m_] := b[x, y, m] = If[x == 0, z^m, b[x - 1, y, m] + If[y > 0, b[x - 1, y - 1, m], 0] + b[x - 1, y + 1, Max[m, y + 1]]];
a[n_] := SeriesCoefficient[b[2n, 0, 0], {z, 0, n}];
a /@ Range[0, 30] (* Jean-François Alcover, May 11 2020, after Alois P. Heinz in A283595 *)

Formula

Recursion: see Maple program.

a(n) = A283595(2n,n).

a(n) ~ sqrt(769 + 2762/sqrt(13)) * (70 + 2*13^(3/2))^n / (3^(3*n+3)*sqrt(3*Pi*n)). - Vaclav Kotesovec, Mar 13 2017