A005220 Number of Dyck paths of knight moves.

1, 0, 1, 0, 3, 2, 12, 14, 54, 86, 274, 528, 1515, 3266, 8854, 20422, 53786, 129368, 336103, 830148, 2145020, 5390580, 13913325, 35378586, 91415954, 234397542, 606983495, 1566013450, 4065765499, 10540066710, 27437831060, 71404804002

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice
• walk

Comments

A Dyck path of knight moves of size n is a path in ZxZ which:

(1) is made only of steps NNE, NEE, SSE and SEE;

(2) starts at (0,0) and ends at (n,0);

(3) never goes strictly below the x-axis. - Gheorghe Coserea, Jan 16 2017

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 0..200
• Jean-Luc Baril and José L. Ramírez, Knight's paths towards Catalan numbers, Univ. Bourgogne Franche-Comté (2022).
• J. Labelle and Y.-N. Yeh, Dyck paths of knight moves, Discrete Applied Math., 24 (1989), 213-221.

Crossrefs

Cf. A285174.

Programs

Mathematica

gf = (1 + 2z + Sqrt[1 - 4z + 4z^2 - 4z^4] - Sqrt[2]*Sqrt[1 - 4z^2 - 2z^4 + (2z + 1)*Sqrt[1 - 4z + 4z^2 - 4z^4]])/(4z^2); CoefficientList[gf + O[z]^32, z] (* Jean-François Alcover, Jul 16 2015 *)

Maxima

a(n):=sum((sum(binomial(j,m-j)*binomial(m+1,j),j,ceiling(m/2),m))*sum((binomial(m+2*k,k)*sum(binomial(k,l)*binomial(k-l,n-m-3*l-k)*(-1)^(n-l-k),l,0,k))/(m+k+1),k,0,n-m),m,0,n); /*  Vladimir Kruchinin, Mar 05 2016 */

PARI

x='x; y='y;
Fxy = x^4*y^4 - x^2*(2*x+1)*y^3 + x*(x^3+2*x+2)*y^2 - (2*x+1)*y + 1;
seq(N) = {
  my(y0 = 1 + O('x^N), y1=0, dFxy=deriv(Fxy, 'y));
  for (k = 1, N,
    y1 = y0 - subst(Fxy, 'y, y0)/subst(dFxy, 'y, y0);
    if (y1 == y0, break()); y0 = y1);
  Vec(y0);
};
seq(32) \\ Gheorghe Coserea, Jan 16 2017

Formula

G.f.: (1+2z+sqrt(1-4z+4z^2-4z^4)-sqrt(2)*sqrt(1-4z^2-2z^4+(2z+1)sqrt(1-4z+4z^2-4z^4)))/[4z^2].

a(n) ~ (2+sqrt(3))*(sqrt(3*(7*sqrt(3)-3)/46)-sqrt((9-5*sqrt(3))/2)) * (1+sqrt(3))^n/(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 13 2013

a(n) = Sum_{m=0..n}((Sum_{j=ceiling(m/2)..m}(binomial(j,m-j)*binomial(m+1,j)))* Sum_{k=0..n-m}((binomial(m+2*k,k)*Sum_{l=0..k}(binomial(k,l)*binomial(k-l,n-m-3*l-k)*(-1)^(n-l-k)))/(m+k+1))). - Vladimir Kruchinin, Mar 05 2016

0 = x^4*y^4 - x^2*(2*x+1)*y^3 + x*(x^3+2*x+2)*y^2 - (2*x+1)*y + 1, where y is the g.f. - Gheorghe Coserea, Jan 16 2017

Extensions

More terms from Emeric Deutsch, Dec 17 2003