A005223 Number of Dyck paths of knight moves.

0, 0, 1, 0, 2, 2, 7, 10, 29, 52, 142, 294, 772, 1732, 4451, 10482, 26715, 64908, 165194, 409720, 1044629, 2627712, 6721492, 17079076, 43853111, 112273270, 289390434, 745262022, 1928015211, 4988699442, 12949776427, 33638741110, 87590340673

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

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• J. Labelle and Y.-N. Yeh, Dyck paths of knight moves, Discrete Applied Math., 24 (1989), 213-221.

Programs

Mathematica

CoefficientList[Series[1-1/(((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)),{z,0,40}],z] (* Harvey P. Dale, Oct 11 2011 *)

Formula

G.f.=1-1/A, 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) ~ (23*sqrt(2*(9-5*sqrt(3))) + sqrt(138*(7*sqrt(3)-3))) * (1+sqrt(3))^n / (184*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 10 2014

A(x) = 1 - 1/A005220(x). - Gheorghe Coserea, Jan 16 2017

Extensions

More terms from Emeric Deutsch, Dec 17 2003