A005223 Number of Dyck paths of knight moves. (Formerly M0369)

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

0,5

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.

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

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 *)

Crossrefs

Sequence in context: A267446 A054226 A000024 * A118680 A278457 A094116

Adjacent sequences: A005220 A005221 A005222 * A005224 A005225 A005226

Keyword

nonn,easy,nice,walk

Author

N. J. A. Sloane

Extensions

More terms from Emeric Deutsch, Dec 17 2003

Status

approved