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

0,2

Links

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

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

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 *)

Crossrefs

Cf. A283595, A283799.

Sequence in context: A006923 A349304 A372507 * A011272 A367062 A366026

Adjacent sequences: A283664 A283665 A283666 * A283668 A283669 A283670

Keyword

nonn

Author

Alois P. Heinz, Mar 13 2017

Status

approved