A005222 Number of Dyck paths of knight moves.

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice
• walk

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.

Programs

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 *)

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

Extensions

More terms from Emeric Deutsch, Dec 17 2003