A005222 Number of Dyck paths of knight moves. (Formerly M3234)

1, 0, 1, 0, 4, 4, 18, 26, 86, 158, 462, 976, 2665, 6082, 16040, 38338, 99536, 244880, 631923, 1583796, 4081939, 10358670, 26728731, 68425494, 176964795, 455967376, 1182454137, 3061954102, 7962768190, 20702327552, 53983118006, 140817757006

Offset

0,5

References

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

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Vaclav Kotesovec, Recurrence (of order 11)

J. Labelle and Y.-N. Yeh, Dyck paths of knight moves, Discrete Applied Math., 24 (1989), 213-221.

Formula

G.f.: A+z^4A^3/(1-zA)^2, where A=(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) ~ c * (1+sqrt(3))^n / n^(3/2), where c = sqrt(341*sqrt(3) - 225 + 3*sqrt(46*(197*sqrt(3) - 22))) / (4*sqrt(23*Pi)) = 0.794168381329... - Vaclav Kotesovec, Feb 29 2016

A(x) = x^2*A005220(x)*A005221(x) + x*A005221(x)^2 + A005220(x). - Gheorghe Coserea, Jan 16 2017

Mathematica

A[x_] = (s*(r-1+x-x^3) + x*(1+x)*(3+r*(x-1) + x*(6*x-5)))/(4*x^3) /. s -> Sqrt[2]*Sqrt[1+r-2*x*(2*x+x^3-r)] /. r -> Sqrt[1-4*x*(1-x+x^3)];
A[x] + O[x]^32 // CoefficientList[#, x]& (* Jean-François Alcover, Mar 26 2017, after Gheorghe Coserea *)

Crossrefs

Sequence in context: A086448 A128090 A119948 * A214166 A214187 A214238

Adjacent sequences: A005219 A005220 A005221 * A005223 A005224 A005225

Keyword

nonn,easy,nice,walk

Author

N. J. A. Sloane

Extensions

More terms from Emeric Deutsch, Dec 17 2003

Status

approved