OFFSET
0,5
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.
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
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy,nice,walk
AUTHOR
EXTENSIONS
More terms from Emeric Deutsch, Dec 17 2003
STATUS
approved